diff --git a/VERSION.txt b/VERSION.txt
index aa70f9b2457..c196b9216c0 100644
--- a/VERSION.txt
+++ b/VERSION.txt
@@ -1 +1 @@
-SageMath version 8.1.beta1, Release Date: 2017-08-04
+SageMath version 8.1.beta2, Release Date: 2017-08-13
diff --git a/build/pkgs/configure/checksums.ini b/build/pkgs/configure/checksums.ini
index e55d7ae64f2..856db7714d1 100644
--- a/build/pkgs/configure/checksums.ini
+++ b/build/pkgs/configure/checksums.ini
@@ -1,4 +1,4 @@
 tarball=configure-VERSION.tar.gz
-sha1=15364988812ace7a0293646fded1f3c0a372ae71
-md5=398aaa4217d841d6353ebf230003902e
-cksum=2273414518
+sha1=0f56bbb0118c834fec6c7adbe6276245f856092b
+md5=993b6a1f19eafc160cad2a67f17aad8b
+cksum=3308302982
diff --git a/build/pkgs/configure/package-version.txt b/build/pkgs/configure/package-version.txt
index 7c022aedd27..0c56bea5944 100644
--- a/build/pkgs/configure/package-version.txt
+++ b/build/pkgs/configure/package-version.txt
@@ -1 +1 @@
-232
+233
diff --git a/build/pkgs/deformation/spkg-check b/build/pkgs/deformation/spkg-check
old mode 100755
new mode 100644
index 35f04fc8862..27cd9419538
--- a/build/pkgs/deformation/spkg-check
+++ b/build/pkgs/deformation/spkg-check
@@ -1,4 +1,2 @@
-#!/usr/bin/env bash
-
 cd src
 $MAKE check
diff --git a/build/pkgs/deformation/spkg-install b/build/pkgs/deformation/spkg-install
old mode 100755
new mode 100644
index b32e5884850..e0985d153bb
--- a/build/pkgs/deformation/spkg-install
+++ b/build/pkgs/deformation/spkg-install
@@ -1,5 +1,3 @@
-#!/usr/bin/env bash
-
 cd src
 
 ./configure --disable-static --prefix="$SAGE_LOCAL" \
diff --git a/build/pkgs/pari/checksums.ini b/build/pkgs/pari/checksums.ini
index be66a9c3b20..9983fa901ed 100644
--- a/build/pkgs/pari/checksums.ini
+++ b/build/pkgs/pari/checksums.ini
@@ -1,4 +1,4 @@
 tarball=pari-VERSION.tar.gz
-sha1=6e5f9b227bb1afe9a8fb67fe1239211714601142
-md5=fec030cfa01fa1815552c7aa362d8bf9
-cksum=3798015133
+sha1=99a8ecd47ee6cdd22098d8585db183e357fb7b71
+md5=ae178b3c984af32f2008d147e901aa76
+cksum=4215303198
diff --git a/build/pkgs/pari/package-version.txt b/build/pkgs/pari/package-version.txt
index 942ba952403..97f22211f46 100644
--- a/build/pkgs/pari/package-version.txt
+++ b/build/pkgs/pari/package-version.txt
@@ -1 +1 @@
-2.9.2.p2
+2.9.3.p0
diff --git a/build/pkgs/pari/patches/fflush.patch b/build/pkgs/pari/patches/fflush.patch
deleted file mode 100644
index d6eb25995a0..00000000000
--- a/build/pkgs/pari/patches/fflush.patch
+++ /dev/null
@@ -1,20 +0,0 @@
-commit c8ddd6e4952a8731f5e99ff17cfa37cb430b47aa
-Author: Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>
-Date:   Tue Mar 28 10:42:54 2017 +0200
-
-    GP was incorrectly issuing fflush(NULL) (From Erik Bray)
-
-diff --git a/src/language/es.c b/src/language/es.c
-index 38f3737..77bcf8f 100644
---- a/src/language/es.c
-+++ b/src/language/es.c
-@@ -532,7 +532,8 @@ _puts_log(const char *s)
-   fputs(s, f);
- }
- static void
--_flush_log(void) { (void)fflush(pari_logfile); }
-+_flush_log(void)
-+{ if (pari_logfile != NULL) (void)fflush(pari_logfile); }
- 
- static void
- normalOutC(char c) { putc(c, pari_outfile); _putc_log(c); }
diff --git a/build/pkgs/pari/patches/polredabs.patch b/build/pkgs/pari/patches/polredabs.patch
deleted file mode 100644
index 98d06b65757..00000000000
--- a/build/pkgs/pari/patches/polredabs.patch
+++ /dev/null
@@ -1,23 +0,0 @@
-From 1420fa40a7d7b43cf2d21c530a0a1d394bac245d Mon Sep 17 00:00:00 2001
-From: Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>
-Date: Sun, 18 Jun 2017 17:30:46 +0200
-Subject: [PATCH] 82- polredabs(non-monic t_POL, 1) => potentially wrong
- transformation map
-
----
- src/basemath/base1.c | 2 +-
- 1 files changed, 1 insertion(+), 1 deletion(-)
-
-diff --git a/src/basemath/base1.c b/src/basemath/base1.c
-index 47c6ba1..9f1c736 100644
---- a/src/basemath/base1.c
-+++ b/src/basemath/base1.c
-@@ -2484,7 +2484,7 @@ polredabs0(GEN x, long flag)
-         GEN w = gel(vw,2);
-         for (i = 1; i < l; i++)
-           w = ZV_union_shallow(w, gel(Z_factor(gel(v,i)),1));
--        nfinit_basic(&S, mkvec2(x,w));
-+        nfinit_basic(&S, mkvec2(S.T0,w));
-       }
-     }
-     v = polredabs_aux(&S, &u);
diff --git a/build/pkgs/polytopes_db/SPKG.txt b/build/pkgs/polytopes_db/SPKG.txt
index 12375de91bd..3f2aae989ff 100644
--- a/build/pkgs/polytopes_db/SPKG.txt
+++ b/build/pkgs/polytopes_db/SPKG.txt
@@ -19,21 +19,3 @@ GPL
 == Dependencies ==
 
 None
-
-== Changelog ==
-
-=== polytopes_db-20120220 (Volker Braun 2012 Feb 20) ===
-* #12553: Added PALP binary data file version. In the future,
-  this will be the only data file format.
-
-=== polytopes_db-20100210.p2 (R. Andrew Ohana, 2012-05-17) ===
-* #13123: Move SAGE_DATA to SAGE_LOCAL/share
-
-=== polytopes_db-20100210.p1 (Keshav Kini, 2012-03-18) ===
-* #12694: Normalize directory structure
-
-=== polytopes_db-20100210 (?, ?) ===
-* Binary files replaced by text ones.
-
-=== polytopes_db-20080430 (?, ?) ===
-* 2008-04-30 Original version.
diff --git a/build/pkgs/polytopes_db/checksums.ini b/build/pkgs/polytopes_db/checksums.ini
index c662abac41b..ceb452697f3 100644
--- a/build/pkgs/polytopes_db/checksums.ini
+++ b/build/pkgs/polytopes_db/checksums.ini
@@ -1,4 +1,4 @@
 tarball=polytopes_db-VERSION.tar.bz2
-sha1=4d83146f9739e1c907b8592f620f97b7eb02bb3d
-md5=99bd633f4dfcdfef33c22b0b1b33919a
-cksum=3316537984
+sha1=6a9d12740588ea5c0efc0e82776d8386e7887b47
+md5=64c5fba9568891a2dbde5d433ce668e2
+cksum=3276110922
diff --git a/build/pkgs/polytopes_db/package-version.txt b/build/pkgs/polytopes_db/package-version.txt
index 2bf1d04f5a3..7c2a5aa9867 100644
--- a/build/pkgs/polytopes_db/package-version.txt
+++ b/build/pkgs/polytopes_db/package-version.txt
@@ -1 +1 @@
-20120220
+20170220
diff --git a/build/pkgs/psutil/patches/cygwin-support.patch b/build/pkgs/psutil/patches/cygwin-support.patch
index 9b9f68854c5..dcdca636c29 100644
--- a/build/pkgs/psutil/patches/cygwin-support.patch
+++ b/build/pkgs/psutil/patches/cygwin-support.patch
@@ -819,7 +819,7 @@ index 0000000..ffba639
 +            raise
 +        # Process name is between parentheses. It can contain spaces and
 +        # other parentheses. This is taken into account by looking for
-+        # the first occurrence of "(" and the last occurence of ")".
++        # the first occurrence of "(" and the last occurrence of ")".
 +        rpar = data.rfind(b')')
 +        name = data[data.find(b'(') + 1:rpar]
 +        fields_after_name = data[rpar + 2:].split()
diff --git a/build/pkgs/rpy2/SPKG.txt b/build/pkgs/rpy2/SPKG.txt
index 37d55804d82..9c78b19257c 100644
--- a/build/pkgs/rpy2/SPKG.txt
+++ b/build/pkgs/rpy2/SPKG.txt
@@ -26,7 +26,7 @@ Website: http://rpy.sourceforge.net/rpy2.html
 
 === rpy2-2.7.4 (Emmanuel Charpentier, December 1st, 2015) ===
  * Trac #19638 : upgrade to allow interaction with Jupyter notebook
- * Updated cygwin.patch to add "|| #defined(__CyGWIN__)" to ALL occurences
+ * Updated cygwin.patch to add "|| #defined(__CyGWIN__)" to ALL occurrences
    of "#defined(Win32) || #defined(Win64)". This should be reviewed by
    someone who knows what (he|she)'s doing with Cygwin (I don't).
  * Removed setup.patch, which as no target now.
diff --git a/src/bin/sage-banner b/src/bin/sage-banner
index 7f5f0fa4873..36b751b7de9 100644
--- a/src/bin/sage-banner
+++ b/src/bin/sage-banner
@@ -1,5 +1,5 @@
 ┌────────────────────────────────────────────────────────────────────┐
-│ SageMath version 8.1.beta1, Release Date: 2017-08-04               │
+│ SageMath version 8.1.beta2, Release Date: 2017-08-13               │
 │ Type "notebook()" for the browser-based notebook interface.        │
 │ Type "help()" for help.                                            │
 └────────────────────────────────────────────────────────────────────┘
diff --git a/src/bin/sage-version.sh b/src/bin/sage-version.sh
index e38c89742a4..a7c9e990f2e 100644
--- a/src/bin/sage-version.sh
+++ b/src/bin/sage-version.sh
@@ -1,4 +1,4 @@
 # Sage version information for shell scripts
 # This file is auto-generated by the sage-update-version script, do not edit!
-SAGE_VERSION='8.1.beta1'
-SAGE_RELEASE_DATE='2017-08-04'
+SAGE_VERSION='8.1.beta2'
+SAGE_RELEASE_DATE='2017-08-13'
diff --git a/src/doc/en/constructions/interface_issues.rst b/src/doc/en/constructions/interface_issues.rst
index 8fc9a788984..40dbad4986f 100644
--- a/src/doc/en/constructions/interface_issues.rst
+++ b/src/doc/en/constructions/interface_issues.rst
@@ -228,7 +228,7 @@ Command-line Sage help
 If you know only part of the name of a Sage command and want to
 know where it occurs in Sage, a new option for 0.10.11 has been
 added to make it easier to hunt it down. Just type
-``sage -grep <string>`` to find all occurences of ``<string>`` in the
+``sage -grep <string>`` to find all occurrences of ``<string>`` in the
 Sage source code. For example,
 
 ::
diff --git a/src/doc/en/developer/coding_basics.rst b/src/doc/en/developer/coding_basics.rst
index fc226c6257a..9e68120e9e0 100644
--- a/src/doc/en/developer/coding_basics.rst
+++ b/src/doc/en/developer/coding_basics.rst
@@ -862,6 +862,18 @@ written.
       5
       7
 
+- **Python3 print:** even if Python2 syntax for print can still be
+  used in your own code for the moment, Python3 syntax for print must
+  be used in Sage code and doctests. If you use an old-style print in
+  doctests, it will raise a SyntaxError::
+
+      sage: print "not like that"
+      Traceback (most recent call last):
+      ...
+      SyntaxError: invalid syntax
+      sage: print("but like this")
+      but like this
+
 - **Split long lines:** You may want to split long lines of code with a
   backslash. Note: this syntax is non-standard and may be removed in the
   future::
diff --git a/src/doc/en/developer/git_trac.rst b/src/doc/en/developer/git_trac.rst
index bd30c04870d..d18160a39de 100644
--- a/src/doc/en/developer/git_trac.rst
+++ b/src/doc/en/developer/git_trac.rst
@@ -486,7 +486,7 @@ The file now looks like this::
     <<<<<<< HEAD
         if i > 1:
             return fibonacci(i-1) * fibonacci(i-2)
-        return i
+        return [0, 1][i]
     =======
         return fibonacci(i-1) + fibonacci(i-2)
     >>>>>>> 41675dfaedbfb89dcff0a47e520be4aa2b6c5d1b
diff --git a/src/doc/en/prep/Quickstarts/NumAnalysis.rst b/src/doc/en/prep/Quickstarts/NumAnalysis.rst
index 70fc163fed8..e3a0d888d07 100644
--- a/src/doc/en/prep/Quickstarts/NumAnalysis.rst
+++ b/src/doc/en/prep/Quickstarts/NumAnalysis.rst
@@ -34,19 +34,16 @@ Basic Analysis
     sage: ring=RealField(3)
 
 To print the actual number (without rounding off the last few imprecise
-digits to only display correct digits), call the ``.str()`` method with
-the option ``truncate=False``.
-
-::
+digits to only display correct digits), call the ``.str()`` method::
 
     sage: print(ring('1').nextabove())
     1.2
 
 ::
 
-    sage: print(ring('1').nextabove().str(truncate=False))
+    sage: print(ring('1').nextabove().str())
     1.2
-    sage: print(ring('1').nextbelow().str(truncate=False))
+    sage: print(ring('1').nextbelow().str())
     0.88
 
 Let's change our precision.
@@ -54,9 +51,9 @@ Let's change our precision.
 ::
 
     sage: ring=RealField(20)
-    sage: print(ring('1').nextabove().str(truncate=False))
+    sage: print(ring('1').nextabove().str())
     1.0000019
-    sage: print(ring('1').nextbelow().str(truncate=False))
+    sage: print(ring('1').nextbelow().str())
     0.99999905
 
 You can also specify the rounding mode.
@@ -68,17 +65,17 @@ You can also specify the rounding mode.
 
 ::
 
-    sage: ring(1/9).str(truncate=False)
+    sage: ring(1/9).str()
     '0.11111116'
 
 ::
 
-    sage: ringup(1/9).str(truncate=False)
+    sage: ringup(1/9).str()
     '0.13'
 
 ::
 
-    sage: ringdown(1/9).str(truncate=False)
+    sage: ringdown(1/9).str()
     '0.10'
 
 ::
diff --git a/src/doc/en/reference/manifolds/diff_map.rst b/src/doc/en/reference/manifolds/diff_map.rst
index 0054c1099ef..385248c0506 100644
--- a/src/doc/en/reference/manifolds/diff_map.rst
+++ b/src/doc/en/reference/manifolds/diff_map.rst
@@ -9,3 +9,5 @@ Differentiable Maps and Curves
    sage/manifolds/differentiable/diff_map
 
    sage/manifolds/differentiable/curve
+   
+   sage/manifolds/differentiable/integrated_curve
diff --git a/src/doc/en/reference/plot3d/threejs_examples/helix.html b/src/doc/en/reference/plot3d/threejs_examples/helix.html
index 4bfa45da48e..9300d18060a 100644
--- a/src/doc/en/reference/plot3d/threejs_examples/helix.html
+++ b/src/doc/en/reference/plot3d/threejs_examples/helix.html
@@ -15,8 +15,8 @@
 
 <body>
 
-<script src=&quot;/Users/Masson/Downloads/GitHub/sage/local/share/threejs/three.min.js&quot;></script>
-<script src=&quot;/Users/Masson/Downloads/GitHub/sage/local/share/threejs/OrbitControls.js&quot;></script>
+<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/build/three.min.js&quot;></script>
+<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/examples/js/controls/OrbitControls.js&quot;></script>
                 
 <script>
 
diff --git a/src/doc/en/reference/plot3d/threejs_examples/spheres.html b/src/doc/en/reference/plot3d/threejs_examples/spheres.html
index 02b8bcbfb06..be2aa61f090 100644
--- a/src/doc/en/reference/plot3d/threejs_examples/spheres.html
+++ b/src/doc/en/reference/plot3d/threejs_examples/spheres.html
@@ -15,8 +15,8 @@
 
 <body>
 
-<script src=&quot;/Users/Masson/Downloads/GitHub/sage/local/share/threejs/three.min.js&quot;></script>
-<script src=&quot;/Users/Masson/Downloads/GitHub/sage/local/share/threejs/OrbitControls.js&quot;></script>
+<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/build/three.min.js&quot;></script>
+<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/examples/js/controls/OrbitControls.js&quot;></script>
                 
 <script>
 
diff --git a/src/doc/en/reference/references/index.rst b/src/doc/en/reference/references/index.rst
index 09cae3e919b..f43fe78a88f 100644
--- a/src/doc/en/reference/references/index.rst
+++ b/src/doc/en/reference/references/index.rst
@@ -85,7 +85,7 @@ REFERENCES:
              *MacMahon's partition analysis: the Omega package*,
              European J. Combin. 22 (2001), no. 7, 887--904.
 
-.. [Ar2006] D. Armstrong. *Generalized noncrossing partitions and
+.. [Ar2006] \D. Armstrong. *Generalized noncrossing partitions and
             combinatorics of Coxeter groups*. Mem. Amer. Math. Soc., 2006.
 
 .. [AR2012] \D. Armstrong and B. Rhoades. "The Shi arrangement and the
@@ -165,6 +165,12 @@ REFERENCES:
              Pseudo-Random Number Generator. *SIAM Journal on
              Computing*, 15(2):364--383, 1986.
 
+.. [BIANCO] \L. Bianco, P. Dell‘Olmo, S. Giordani
+            An Optimal Algorithm to Find the Jump Number of Partially Ordered Sets
+            Computational Optimization and Applications,
+            1997, Volume 8, Issue 2, pp 197--210,
+            :doi:`10.1023/A:1008625405476`
+
 .. [BC1977] \R. E. Bixby, W. H. Cunningham, Matroids, Graphs, and
             3-Connectivity. In Graph theory and related topics
             (Proc. Conf., Univ. Waterloo, Waterloo, ON, 1977), 91-103
@@ -189,6 +195,11 @@ REFERENCES:
                  203–218. Springer, 2010. pre-print available at
                  http://eprint.iacr.org/2010/313.pdf
 
+.. [BF1999] Thomas Britz, Sergey Fomin,
+            *Finite posets and Ferrers shapes*,
+            Advances in Mathematics 158, pp. 86-127 (2001),
+            :arxiv:`math/9912126` (the arXiv version has fewer errors).
+
 .. [BDP2013] Thomas Brüstle, Grégoire Dupont, Matthieu Pérotin
    *On Maximal Green Sequences*
    :arxiv:`1205.2050`
@@ -238,10 +249,15 @@ REFERENCES:
 .. [BK1992] \U. Brehm and W. Kuhnel, "15-vertex triangulations of an
             8-manifold", Math. Annalen 294 (1992), no. 1, 167-193.
 
-.. [BK2001] W. Bruns and R. Koch, Computing the integral closure of an
+.. [BK2001] \W. Bruns and R. Koch, Computing the integral closure of an
             affine semigroup. Uni. Iaggelonicae Acta Math. 39, (2001),
             59-70
 
+.. [Bj1980] Anders Björner,
+            *Shellable and Cohen-Macaulay partially ordered sets*,
+            Trans. Amer. Math. Soc. 260 (1980), 159-183,
+            :doi:`10.1090/S0002-9947-1980-0570784-2`
+
 .. [BL2000] Anders Björner and Frank H. Lutz, "Simplicial manifolds,
             bistellar flips and a 16-vertex triangulation of the
             Poincaré homology 3-sphere", Experiment. Math. 9 (2000),
@@ -374,7 +390,7 @@ REFERENCES:
              Secret. Cryptology ePrint Archive, Report 2013/164. 2013.
              2013/164. http://eprint.iacr.org/2013/164
 
-.. [CGMRV16] A. Conte, R. Grossi, A. Marino, R. Rizzi, L. Versari,
+.. [CGMRV16] \A. Conte, R. Grossi, A. Marino, R. Rizzi, L. Versari,
              "Directing Road Networks by Listing Strong Orientations.",
              Combinatorial Algorithms, Proceedings of 27th International Workshop,
              IWOCA 2016, August 17-19, 2016, pages 83--95.
@@ -398,6 +414,11 @@ REFERENCES:
 .. [ChenDB] Eric Chen, Online database of two-weight codes,
             http://moodle.tec.hkr.se/~chen/research/2-weight-codes/search.php
 
+.. [CHK2001] Keith D. Cooper, Timothy J. Harvey and Ken Kennedy. *A
+             Simple, Fast Dominance Algorithm*, Software practice and
+             Experience, 4:1-10 (2001).
+             http://www.hipersoft.rice.edu/grads/publications/dom14.pdf
+
 .. [CK1999] David A. Cox and Sheldon Katz. *Mirror symmetry and
             algebraic geometry*, volume 68 of *Mathematical Surveys
             and Monographs*. American Mathematical Society,
@@ -523,7 +544,7 @@ REFERENCES:
              *Introduction to Lattices and Order*,
              Cambridge University Press, 1997.
 
-.. [Dec1998] W. Decker and T. de Jong. Groebner Bases and Invariant
+.. [Dec1998] \W. Decker and T. de Jong. Groebner Bases and Invariant
              Theory in Groebner Bases and Applications. London
              Mathematical Society Lecture Note Series No. 251. (1998)
              61--89.
@@ -678,6 +699,10 @@ REFERENCES:
 .. [EZ1950] \S. Eilenberg and J. Zilber, "Semi-Simplicial Complexes
             and Singular Homology", Ann. Math. (2) 51 (1950), 499-513.
 
+.. [EPW14] Ben Elias, Nicholas Proudfoot, and Max Wakefield.
+           *The Kazhdan-Lusztig polynomial of a matroid*. 2014.
+           :arxiv:`1412.7408`.
+
 .. _ref-F:
 
 **F**
@@ -687,6 +712,12 @@ REFERENCES:
             Computational Geometry 18 (1997),
             257-267. http://page.math.tu-berlin.de/~felsner/Paper/numarr.pdf
 
+.. [FT00] Stefan Felsner, William T. Trotter,
+          Dimension, Graph and Hypergraph Coloring,
+          Order,
+          2000, Volume 17, Issue 2, pp 167-177,
+          http://link.springer.com/article/10.1023%2FA%3A1006429830221
+
 .. [Fe2012] Hans L. Fetter, "A Polyhedron Full of Surprises",
             Mathematics Magazine 85 (2012), no. 5, 334-342.
 
@@ -759,6 +790,11 @@ REFERENCES:
              operations on finite simplicial complexes* in Homology,
              Homotopy and Applications 5 (2003), 83-93.
 
+.. [Ge2005] Loukas Georgiadis, *Linear-Time Algorithms for Dominators
+            and Related Problems*, PhD thesis, Princetown University,
+            TR-737-05, (2005).
+            ftp://ftp.cs.princeton.edu/reports/2005/737.pdf
+
 .. [GG2012] Jim Geelen and Bert Gerards, Characterizing graphic
             matroids by a system of linear equations,
             submitted, 2012. Preprint:
@@ -768,7 +804,7 @@ REFERENCES:
              Riemann surfaces and Dessins d'enfant*, (2011)
              London Mathematical Society, Student Text 79.
 
-.. [GGOR2003] V. Ginzberg, N. Guay, E. Opdam, R. Rouquier.
+.. [GGOR2003] \V. Ginzberg, N. Guay, E. Opdam, R. Rouquier.
               *On the category `\mathcal{O}` for rational Cherednik algebras*.
               Invent. Math. **154** (2003). :arxiv:`math/0212036`.
 
@@ -818,6 +854,10 @@ REFERENCES:
             birational rowmotion
             I*. http://web.mit.edu/~darij/www/algebra/skeletal.pdf
 
+.. [Grinb2016a] Darij Grinberg,
+                *Double posets and the antipode of QSym*,
+                :arxiv:`1509.08355v2`.
+
 .. [GrS1967] Grunbaum and Sreedharan, "An enumeration of simplicial
              4-polytopes with 8 vertices", J. Comb. Th. 2,
              437-465 (1967)
@@ -931,6 +971,12 @@ REFERENCES:
 
 **I**
 
+.. [ILS2012] Giuseppe F. Italiano, Luigi Laura, and Federico
+             Santaroni. *Finding strong bridges and strong
+             articulation points in linear time*. Theoretical Computer
+             Science, 447, 74–84 (2012).
+             :doi:`10.1016/j.tcs.2011.11.011`
+
 .. [IR1990] \K. Ireland and M. Rosen, *A Classical Introduction to
             Modern Number Theory*, Springer-Verlag, GTM volume
             84, 1990.
@@ -965,6 +1011,10 @@ REFERENCES:
 .. [Jon2005] \V. Jones, The Jones
              Polynomial, 2005. https://math.berkeley.edu/~vfr/jones.pdf
 
+.. [JRJ94] Jourdan, Guy-Vincent; Rampon, Jean-Xavier; Jard, Claude
+           (1994), "Computing on-line the lattice of maximal antichains
+           of posets", Order 11 (3) p. 197-210, :doi:`10.1007/BF02115811`
+
 .. [Joy2004] \D. Joyner, Toric codes over finite fields, Applicable
              Algebra in Engineering, Communication and Computing, 15,
              (2004), p. 63-79.
@@ -1205,7 +1255,7 @@ REFERENCES:
              modular functions*, Int. Math. Res. Not 2007 (050).
              `Arxiv preprint <http://uk.arxiv.org/abs/math/0701168>`_.
 
-.. [Lot2005] M. Lothaire, *Applied combinatorics on
+.. [Lot2005] \M. Lothaire, *Applied combinatorics on
              words*. Cambridge University Press (2005).
 
 .. [LP2011] Richard Lindner and Chris Peikert. Better key sizes (and
@@ -1460,7 +1510,7 @@ REFERENCES:
 .. [Prototype_pattern] Prototype pattern,
                        :wikipedia:`Prototype_pattern`
 
-.. [PS2011] R. Pollack, and G. Stevens.  *Overconvergent modular
+.. [PS2011] \R. Pollack, and G. Stevens.  *Overconvergent modular
             symbols and p-adic L-functions.* Annales scientifiques de
             l'Ecole normale superieure.
             Vol. 44. No. 1. Elsevier, 2011.
@@ -1476,6 +1526,12 @@ REFERENCES:
 .. [PZ2008] \J. H. Palmieri and J. J. Zhang, "Commutators in the
             Steenrod algebra," New York J. Math. 19 (2013), 23-37.
 
+.. [Propp1997] James Propp,
+               *Generating Random Elements of Finite Distributive Lattices*,
+               Electron. J. Combin. 4 (1997), no. 2, The Wilf Festschrift volume,
+               Research Paper 15.
+               http://www.combinatorics.org/ojs/index.php/eljc/article/view/v4i2r15
+
 .. _ref-Q:
 .. _ref-R:
 
@@ -1625,12 +1681,27 @@ REFERENCES:
             Electronic Journal of Combinatorics. **19(2)**. 2012.
             :arXiv:`1104.2359`
 
+.. [St1986] Richard Stanley. *Two poset polytopes*,
+            Discrete Comput. Geom. (1986), :doi:`10.1007/BF02187680`
+
 .. [Sta2007] Stanley, Richard: *Hyperplane Arrangements*, Geometric
              Combinatorics (E. Miller, V. Reiner, and B. Sturmfels,
              eds.), IAS/Park City Mathematics Series, vol. 13,
              American Mathematical Society, Providence, RI, 2007,
              pp. 389-496.
 
+.. [EnumComb1] Stanley, Richard P.
+               *Enumerative Combinatorics, volume 1*,
+               Second Edition,
+               Cambridge University Press (2011).
+               http://math.mit.edu/~rstan/ec/ec1/
+
+.. [Stan2009] Richard Stanley,
+              *Promotion and evacuation*,
+              Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of
+              Anders Björner,
+              Research Paper 9, 24 pp.
+
 .. [Ste2003] John R. Stembridge, A local characterization of
              simply-laced crystals, Transactions of the American
              Mathematical Society, Vol. 355, No. 12 (Dec., 2003),
@@ -1660,7 +1731,7 @@ REFERENCES:
 .. [Stu1993] \B. Sturmfels, Algorithms in invariant theory, Springer-Verlag,
              1993.
 
-.. [STW2016] C. Stump, H. Thomas, N. Williams. *Cataland II*, in
+.. [STW2016] \C. Stump, H. Thomas, N. Williams. *Cataland II*, in
              preparation, 2016.
 
 .. [sudoku:escargot]  "Al Escargot", due to Arto Inkala,
@@ -1706,11 +1777,11 @@ REFERENCES:
              Jacobi elliptic functions for real
              argument". 1997. https://researchspace.auckland.ac.nz/bitstream/handle/2292/5042/390.pdf.
 
-.. [TIDES] A. Abad, R. Barrio, F. Blesa, M. Rodriguez. TIDES tutorial:
+.. [TIDES] \A. Abad, R. Barrio, F. Blesa, M. Rodriguez. TIDES tutorial:
            Integrating ODEs by using the Taylor Series Method
            (http://www.unizar.es/acz/05Publicaciones/Monografias/MonografiasPublicadas/Monografia36/IndMonogr36.htm)
 
-.. [TOPCOM] J. Rambau, TOPCOM
+.. [TOPCOM] \J. Rambau, TOPCOM
             <http://www.rambau.wm.uni-bayreuth.de/TOPCOM/>.
 
 .. [TW1980] \A.D. Thomas and G.V. Wood, Group Tables (Exeter: Shiva
@@ -1833,10 +1904,10 @@ REFERENCES:
              constrained optimization. ACM Transactions on
              Mathematical Software, Vol 23, Num. 4, pp.550--560, 1997.
 
-.. [Zie1998] G. M. Ziegler. *Shelling polyhedral 3-balls and
+.. [Zie1998] \G. M. Ziegler. *Shelling polyhedral 3-balls and
              4-polytopes*. Discrete Comput. Geom. 19 (1998), 159-174.
 
-.. [Zie2007] G. M. Ziegler. *Lectures on polytopes*, Volume
+.. [Zie2007] \G. M. Ziegler. *Lectures on polytopes*, Volume
              152 of Graduate Texts in Mathematics, 7th printing of 1st edition, Springer, 2007.
 
 .. [Zor2008] \A. Zorich "Explicit Jenkins-Strebel representatives of
diff --git a/src/ext/pari/dokchitser/computel.gp b/src/ext/pari/dokchitser/computel.gp
index 55c50602c27..85f8ce042a0 100644
--- a/src/ext/pari/dokchitser/computel.gp
+++ b/src/ext/pari/dokchitser/computel.gp
@@ -1,4 +1,4 @@
-/************* ComputeL v1.3.3, 2001-2013, (c) Tim Dokchitser ************/
+/************* ComputeL v1.3.4, 2001-2016, (c) Tim Dokchitser ************/
 /**************** computing special values of L-functions ****************/
 /* arXiv.org/abs/math.NT/0207280, Exper. Math. 13 (2004), no. 2, 137-150 */
 /****** Questions/comments welcome! -> tim.dokchitser@bristol.ac.uk ******/
@@ -353,7 +353,7 @@ cflength(cutoff=1.2)=
     t  = if(t1,(t1+t2)\2,t2);
     tt = t/cutoff/vA;
     res = coefgrow(t) * asympconstf*exp(-d*tt^(2/d))*tt^expdifff;
-    if (t1,if(res>err,t1=t,t2=t),if(res<err,t1=t2/2,t2*=2));
+    if (t1,if(abs(res)>err,t1=t,t2=t),if(abs(res)<err,t1=t2/2,t2*=2));
   );
   ceil(t2)
 }
diff --git a/src/ext/pari/dokchitser/computel.gp.template b/src/ext/pari/dokchitser/computel.gp.template
index 2906481f917..24beeb307af 100644
--- a/src/ext/pari/dokchitser/computel.gp.template
+++ b/src/ext/pari/dokchitser/computel.gp.template
@@ -1,4 +1,4 @@
-/************* ComputeL v1.3.3, 2001-2013, (c) Tim Dokchitser ************/
+/************* ComputeL v1.3.4, 2001-2016, (c) Tim Dokchitser ************/
 /**************** computing special values of L-functions ****************/
 /* arXiv.org/abs/math.NT/0207280, Exper. Math. 13 (2004), no. 2, 137-150 */
 /****** Questions/comments welcome! -> tim.dokchitser@bristol.ac.uk ******/
@@ -353,7 +353,7 @@ cflength_$i(cutoff=1.2)=
     t  = if(t1,(t1+t2)\2,t2);
     tt = t/cutoff/vA_$i;
     res = coefgrow_$i(t) * asympconstf_$i*exp(-d*tt^(2/d))*tt^expdifff_$i;
-    if (t1,if(res>err,t1=t,t2=t),if(res<err,t1=t2/2,t2*=2));
+    if (t1,if(abs(res)>err,t1=t,t2=t),if(abs(res)<err,t1=t2/2,t2*=2));
   );
   ceil(t2)
 }
diff --git a/src/ext/pari/dokchitser/ex-chqua b/src/ext/pari/dokchitser/ex-chqua
index 445abfb459c..d969ef77301 100644
--- a/src/ext/pari/dokchitser/ex-chqua
+++ b/src/ext/pari/dokchitser/ex-chqua
@@ -7,7 +7,7 @@ read("computel");                 \\ read the ComputeL package
                                   \\ and set the default values
 default(realprecision,28);        \\ set working precision; used throughout
 
-P = 37;                     \\ May change this to to any other odd prime
+P = 37;                     \\ May change this to any other odd prime
                             \\ Chi = Legendre character modulo this P
 
                             \\ initialize L-function parameters
diff --git a/src/sage/algebras/cluster_algebra.py b/src/sage/algebras/cluster_algebra.py
index d0f4a1f3dc2..b37425d5be5 100644
--- a/src/sage/algebras/cluster_algebra.py
+++ b/src/sage/algebras/cluster_algebra.py
@@ -50,7 +50,7 @@
 and coefficients.
 
 :class:`ClusterAlgebraElement` is a thin wrapper around
-:class:`sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_generic`
+:class:`sage.rings.polynomial.laurent_polynomial.LaurentPolynomial`
 providing all the functions specific to cluster variables.
 Elemets of a cluster algebra with principal coefficients have special methods
 and these are grouped in the subclass :class:`PrincipalClusterAlgebraElement`.
diff --git a/src/sage/algebras/free_algebra.py b/src/sage/algebras/free_algebra.py
index 1caf0c1a9ce..4e1f5b02495 100644
--- a/src/sage/algebras/free_algebra.py
+++ b/src/sage/algebras/free_algebra.py
@@ -404,7 +404,7 @@ def __init__(self, R, n, names):
             sage: F.<x,y,z> = FreeAlgebra(QQ, 3); F # indirect doctet
             Free Algebra on 3 generators (x, y, z) over Rational Field
 
-        TEST:
+        TESTS:
 
         Note that the following is *not* the recommended way to create
         a free algebra::
diff --git a/src/sage/algebras/letterplace/free_algebra_element_letterplace.pxd b/src/sage/algebras/letterplace/free_algebra_element_letterplace.pxd
index fa7df069e42..d22fe4e9a40 100644
--- a/src/sage/algebras/letterplace/free_algebra_element_letterplace.pxd
+++ b/src/sage/algebras/letterplace/free_algebra_element_letterplace.pxd
@@ -13,3 +13,5 @@ from sage.algebras.letterplace.free_algebra_letterplace cimport FreeAlgebra_lett
 
 cdef class FreeAlgebraElement_letterplace(AlgebraElement):
     cdef MPolynomial_libsingular _poly
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
diff --git a/src/sage/algebras/letterplace/free_algebra_letterplace.pyx b/src/sage/algebras/letterplace/free_algebra_letterplace.pyx
index be442aa7baa..ed8a70f5932 100644
--- a/src/sage/algebras/letterplace/free_algebra_letterplace.pyx
+++ b/src/sage/algebras/letterplace/free_algebra_letterplace.pyx
@@ -140,7 +140,7 @@ cdef MPolynomialRing_libsingular make_letterplace_ring(base_ring,blocks):
     variable names of the `n`-th block (`n>0`) ending with
     ``"_%d"%n``.
 
-    TEST:
+    TESTS:
 
     Note that, since the algebras are cached, we need to choose
     a different base ring, since other doctests could have a
@@ -728,7 +728,7 @@ cdef class FreeAlgebra_letterplace(Algebra):
           generators are equal, and the base ring of ``R`` coerces
           into the base ring of self.
 
-        TEST:
+        TESTS:
 
         Coercion from the base ring::
 
@@ -815,7 +815,7 @@ cdef class FreeAlgebra_letterplace(Algebra):
           This is forwarded to the initialisation of
           :class:`~sage.algebras.letterplace.free_algebra_element_letterplace.FreeAlgebraElement_letterplace`.
 
-        TEST:
+        TESTS:
 
         This method applied to the dictionary of any element must
         return the same element. This must hold true even if the
diff --git a/src/sage/algebras/lie_algebras/poincare_birkhoff_witt.py b/src/sage/algebras/lie_algebras/poincare_birkhoff_witt.py
index b300ece8281..65986ef87df 100644
--- a/src/sage/algebras/lie_algebras/poincare_birkhoff_witt.py
+++ b/src/sage/algebras/lie_algebras/poincare_birkhoff_witt.py
@@ -267,14 +267,34 @@ def _coerce_map_from_(self, R):
              - 4*PBW[-alpha[1]]*PBW[alpha[1]]
              + PBW[alphacheck[1]]^2
              - 2*PBW[alphacheck[1]]
+
+        TESTS:
+
+        Check that we can take the preimage (:trac:`23375`)::
+
+            sage: L = lie_algebras.cross_product(QQ)
+            sage: pbw = L.pbw_basis()
+            sage: L(pbw(L.an_element()))
+            X + Y + Z
+            sage: L(pbw(L.an_element())) == L.an_element()
+            True
+            sage: L(prod(pbw.gens()))
+            Traceback (most recent call last):
+            ValueError: PBW['X']*PBW['Y']*PBW['Z'] is not in the image
+            sage: L(pbw.one())
+            Traceback (most recent call last):
+            ...
+            ValueError: 1 is not in the image
         """
         if R == self._g:
             # Make this into the lift map
             I = self._indices
-            basis_function = lambda x: self.monomial(I.gen(x))
+            def basis_function(x): return self.monomial(I.gen(x))
+            def inv_supp(m): return None if m.length() != 1 else m.leading_support()
             # TODO: this diagonal, but with a smaller indexing set...
             return self._g.module_morphism(basis_function, codomain=self,
-                                           triangular='upper', unitriangular=True)
+                                           triangular='upper', unitriangular=True,
+                                           inverse_on_support=inv_supp)
 
         if isinstance(R, PoincareBirkhoffWittBasis) and self._g == R._g:
             I = self._indices
diff --git a/src/sage/algebras/lie_algebras/structure_coefficients.py b/src/sage/algebras/lie_algebras/structure_coefficients.py
index e01d8489d49..6df420eec19 100644
--- a/src/sage/algebras/lie_algebras/structure_coefficients.py
+++ b/src/sage/algebras/lie_algebras/structure_coefficients.py
@@ -331,6 +331,22 @@ def monomial(self, k):
         """
         return self.element_class(self, self._M.basis()[self._index_to_pos[k]])
 
+    def term(self, k, c=None):
+        """
+        Return the term indexed by ``i`` with coefficient ``c``.
+
+        EXAMPLES::
+
+            sage: L.<x,y,z> = LieAlgebra(QQ, {('x','y'): {'z':1}})
+            sage: L.term('x', 4)
+            4*x
+        """
+        if c is None:
+            c = self.base_ring().one()
+        else:
+            c = self.base_ring()(c)
+        return self.element_class(self, c * self._M.basis()[self._index_to_pos[k]])
+
     def from_vector(self, v):
         """
         Return an element of ``self`` from the vector ``v``.
diff --git a/src/sage/calculus/interpolators.pyx b/src/sage/calculus/interpolators.pyx
index 36ee8df1f1e..cef315dddb9 100644
--- a/src/sage/calculus/interpolators.pyx
+++ b/src/sage/calculus/interpolators.pyx
@@ -22,6 +22,7 @@ Development supported by NSF award No. 0702939.
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 import numpy as np
 cimport numpy as np
diff --git a/src/sage/calculus/riemann.pyx b/src/sage/calculus/riemann.pyx
index f0ebd6be2c9..01248935392 100644
--- a/src/sage/calculus/riemann.pyx
+++ b/src/sage/calculus/riemann.pyx
@@ -24,7 +24,7 @@ Development supported by NSF award No. 0702939.
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from cysignals.signals cimport sig_on, sig_off
 
diff --git a/src/sage/categories/category_with_axiom.py b/src/sage/categories/category_with_axiom.py
index a39f5353849..318284666bf 100644
--- a/src/sage/categories/category_with_axiom.py
+++ b/src/sage/categories/category_with_axiom.py
@@ -926,7 +926,7 @@ def _(): return LazyImport('sage.categories.rngs', 'Rngs', at_startup=True)
 .. TOPIC:: Design discussion
 
     Supporting similar deduction rules will be an important feature in
-    the future, with quite a few occurences already implemented in
+    the future, with quite a few occurrences already implemented in
     upcoming tickets. For the time being though there is a single
     occurrence of this idiom outside of the tests. So this would be an
     easy thing to refactor after :trac:`10963` if a better idiom is
diff --git a/src/sage/categories/finite_complex_reflection_groups.py b/src/sage/categories/finite_complex_reflection_groups.py
index d276361242d..05ffa06c421 100644
--- a/src/sage/categories/finite_complex_reflection_groups.py
+++ b/src/sage/categories/finite_complex_reflection_groups.py
@@ -466,7 +466,7 @@ def is_real(self):
             irreducible complex reflection group is real if and only
             if `2` is a degree of ``self`` with multiplicity one.
             Hence, in general we just need to compare the number of
-            occurences of `2` as degree of ``self`` and the number of
+            occurrences of `2` as degree of ``self`` and the number of
             irreducible components.
 
             EXAMPLES::
diff --git a/src/sage/categories/finite_monoids.py b/src/sage/categories/finite_monoids.py
index f0a458d1df0..2e5986b2101 100644
--- a/src/sage/categories/finite_monoids.py
+++ b/src/sage/categories/finite_monoids.py
@@ -77,7 +77,7 @@ def nerve(self):
 
             The `k`-simplices of the nerve are named after the chains
             of `k` non-unit elements to be multiplied. The group
-            `\Sigma_2` has two elements, writen ``()`` (the identity
+            `\Sigma_2` has two elements, written ``()`` (the identity
             element) and ``(1,2)`` in Sage. So the 1-cells and 2-cells
             in `B\Sigma_2` are::
 
diff --git a/src/sage/categories/finite_posets.py b/src/sage/categories/finite_posets.py
index 2d6f1c1d534..af85790a6d4 100644
--- a/src/sage/categories/finite_posets.py
+++ b/src/sage/categories/finite_posets.py
@@ -51,49 +51,70 @@ class ParentMethods:
 
         def is_lattice(self):
             r"""
-            Returns whether this poset is both a meet and a join semilattice.
+            Return whether the poset is a lattice.
+
+            A poset is a lattice if all pairs of elements have
+            both a least upper bound ("join") and a greatest lower bound
+            ("meet") in the poset.
 
             EXAMPLES::
 
-                sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]])
+                sage: P = Poset([[1, 3, 2], [4], [4, 5, 6], [6], [7], [7], [7], []])
                 sage: P.is_lattice()
                 True
 
-                sage: P = Poset([[1,2],[3],[3],[]])
+                sage: P = Poset([[1, 2], [3], [3], []])
                 sage: P.is_lattice()
                 True
 
-                sage: P = Poset({0:[2,3],1:[2,3]})
+                sage: P = Poset({0: [2, 3], 1: [2, 3]})
                 sage: P.is_lattice()
                 False
+
+                sage: P = Poset({1: [2, 3, 4], 2: [5, 6], 3: [5, 7], 4: [6, 7], 5: [8, 9],
+                ....:            6: [8, 10], 7: [9, 10], 8: [11], 9: [11], 10: [11]})
+                sage: P.is_lattice()
+                False
+
+            TESTS::
+
+                sage: P = Poset()
+                sage: P.is_lattice()
+                True
+
+            .. SEEALSO::
+
+                - Weaker properties: :meth:`~sage.combinat.posets.posets.FinitePoset.is_join_semilattice`,
+                  :meth:`~sage.combinat.posets.posets.FinitePoset.is_meet_semilattice`
             """
             return (self.cardinality() == 0 or
                      (self.has_bottom() and self.is_join_semilattice()))
 
         def is_selfdual(self):
             r"""
-            Returns whether this poset is *self-dual*, that is
-            isomorphic to its dual poset.
+            Return whether the poset is *self-dual*.
+
+            A poset is self-dual if it is isomorphic to its dual poset.
 
             EXAMPLES::
 
-                sage: P = Poset(([1,2,3],[[1,3],[2,3]]),cover_relations=True)
+                sage: P = Poset({1: [3, 4], 2: [3, 4]})
                 sage: P.is_selfdual()
-                False
+                True
 
-                sage: P = Poset(([1,2,3,4],[[1,3],[1,4],[2,3],[2,4]]),cover_relations=True)
+                sage: P = Poset({1: [2, 3]})
                 sage: P.is_selfdual()
-                True
+                False
 
             TESTS::
 
-                sage: P = Poset( {} )
+                sage: P = Poset()
                 sage: P.is_selfdual()
                 True
 
             .. SEEALSO::
 
-                - Stronger properties: :meth:`~sage.combinat.posets.lattices.FiniteLattice.is_orthocomplemented`
+                - Stronger properties: :meth:`~sage.combinat.posets.lattices.FiniteLatticePoset.is_orthocomplemented` (for lattices)
             """
             # Two quick checks before full isomorphic test.
             if sorted(self._hasse_diagram.in_degree()) != sorted(self._hasse_diagram.out_degree()):
diff --git a/src/sage/categories/map.pyx b/src/sage/categories/map.pyx
index 64827ea50e2..fddb209b25c 100644
--- a/src/sage/categories/map.pyx
+++ b/src/sage/categories/map.pyx
@@ -699,7 +699,7 @@ cdef class Map(Element):
             sage: phi(I)
             Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field
 
-        TEST:
+        TESTS:
 
         We test that the map can be applied to something that converts
         (but not coerces) into the domain and can *not* be dealt with
@@ -1002,7 +1002,7 @@ cdef class Map(Element):
             sage: phi_xz.category_for()
             Category of monoids
 
-        TEST:
+        TESTS:
 
         This illustrates that it is not tested whether the maps can actually
         be composed, i.e., whether codomain and domain match.
diff --git a/src/sage/categories/morphism.pyx b/src/sage/categories/morphism.pyx
index 1b25b9a53e6..7ffdfa67e54 100644
--- a/src/sage/categories/morphism.pyx
+++ b/src/sage/categories/morphism.pyx
@@ -19,14 +19,14 @@ AUTHORS:
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from cpython.object cimport *
 from sage.misc.constant_function import ConstantFunction
 
 import operator
 
-import homset
+
 
 from sage.structure.element cimport Element
 from sage.structure.richcmp cimport richcmp_not_equal, rich_to_bool
@@ -407,8 +407,9 @@ cdef class CallMorphism(Morphism):
 cdef class IdentityMorphism(Morphism):
 
     def __init__(self, parent):
-        if not isinstance(parent, homset.Homset):
-            parent = homset.Hom(parent, parent)
+        from .homset import Homset, Hom
+        if not isinstance(parent, Homset):
+            parent = Hom(parent, parent)
         Morphism.__init__(self, parent)
 
     def _repr_type(self):
@@ -444,6 +445,28 @@ cdef class IdentityMorphism(Morphism):
     def __invert__(self):
         return self
 
+    def is_surjective(self):
+        r"""
+        Return whether this morphism is surjective.
+
+        EXAMPLES::
+
+            sage: Hom(ZZ, ZZ).identity().is_surjective()
+            True
+        """
+        return True
+
+    def is_injective(self):
+        r"""
+        Return whether this morphism is injective.
+
+        EXAMPLES::
+
+            sage: Hom(ZZ, ZZ).identity().is_injective()
+            True
+        """
+        return True
+
 cdef class SetMorphism(Morphism):
     def __init__(self, parent, function):
         """
diff --git a/src/sage/categories/pushout.py b/src/sage/categories/pushout.py
index f814b797feb..8d2476ca03f 100644
--- a/src/sage/categories/pushout.py
+++ b/src/sage/categories/pushout.py
@@ -1564,7 +1564,7 @@ def _apply_functor(self, R):
         """
         Apply the functor to an object of ``self``'s domain.
 
-        TEST:
+        TESTS:
 
         The following is a test against a bug discussed at :trac:`8800`::
 
@@ -2371,7 +2371,7 @@ def __init__(self, p, prec, extras=None):
                     raise ValueError("completion type must be one of %s"%(", ".join(self._real_types)))
             else:
                 if self.type not in self._dvr_types:
-                    raise ValueError("completion type must be one of %s"%(", ".join(self._dvr_types)))
+                    raise ValueError("completion type must be one of %s"%(", ".join(self._dvr_types[1:])))
 
     def _repr_(self):
         """
@@ -2473,7 +2473,7 @@ def __ne__(self, other):
         return not (self == other)
 
     _real_types = ['Interval','Ball','MPFR','RDF','RLF']
-    _dvr_types = [None, 'fixed-mod','capped-abs','capped-rel','lazy']
+    _dvr_types = [None, 'fixed-mod','floating-point','capped-abs','capped-rel','lazy']
 
     def merge(self, other):
         """
@@ -2557,9 +2557,9 @@ def merge(self, other):
                 return CompletionFunctor(self.p, new_prec, {'type':new_type, 'sci_not':new_scinot, 'rnd':new_rnd})
             else:
                 new_type = self._dvr_types[min(self._dvr_types.index(self.type), self._dvr_types.index(other.type))]
-                if new_type == 'fixed-mod':
-                    if self.type != 'fixed-mod' or other.type != 'fixed-mod':
-                        return None # no coercion into fixed-mod
+                if new_type in ('fixed-mod', 'floating-point'):
+                    if self.type != other.type:
+                        return None # no coercion into fixed-mod or floating-point
                     new_prec = min(self.prec, other.prec)
                 else:
                     new_prec = max(self.prec, other.prec) # since elements track their own precision, we don't want to truncate them
@@ -2585,7 +2585,7 @@ def commutes(self,other):
             sage: F1.commutes(F2)
             True
 
-        TEST:
+        TESTS:
 
         The fraction field ``R`` in the example below has no completion
         method. But completion commutes with the fraction field functor,
diff --git a/src/sage/categories/tutorial.py b/src/sage/categories/tutorial.py
index 08074038360..4e577698517 100644
--- a/src/sage/categories/tutorial.py
+++ b/src/sage/categories/tutorial.py
@@ -19,7 +19,7 @@
 :class:`sage.categories.examples.finite_semigroups.LeftRegularBand`.
 
 Copy-paste this code into, say, a cell of the notebook, and replace
-every occurence of ``FiniteSemigroups().example(...)`` in the
+every occurrence of ``FiniteSemigroups().example(...)`` in the
 documentation by ``LeftRegularBand``. This will be equivalent to::
 
     sage: from sage.categories.examples.finite_semigroups import LeftRegularBand
diff --git a/src/sage/coding/guava.py b/src/sage/coding/guava.py
index 0d1a7265881..45d77ad2ca9 100644
--- a/src/sage/coding/guava.py
+++ b/src/sage/coding/guava.py
@@ -16,7 +16,7 @@
 - David Joyner (2008-03): removed QR, XQR, cyclic and ReedSolomon codes
 
 - David Joyner (2009-05): added "optional package" comments, fixed some
-  docstrings to to be sphinx compatible
+  docstrings to be sphinx compatible
 
 
 REFERENCES:
diff --git a/src/sage/combinat/crystals/pbw_crystal.py b/src/sage/combinat/crystals/pbw_crystal.py
index 42467d2e85c..a1ac25af0f7 100644
--- a/src/sage/combinat/crystals/pbw_crystal.py
+++ b/src/sage/combinat/crystals/pbw_crystal.py
@@ -8,7 +8,7 @@
 
 .. SEEALSO::
 
-    For infromation on PBW datum, see
+    For information on PBW datum, see
     :ref:`sage.combinat.crystals.pbw_datum`.
 """
 
diff --git a/src/sage/combinat/e_one_star.py b/src/sage/combinat/e_one_star.py
index 5857f5bd560..4c3d78d3316 100644
--- a/src/sage/combinat/e_one_star.py
+++ b/src/sage/combinat/e_one_star.py
@@ -273,7 +273,7 @@ def __init__(self, v, t, color=None):
             sage: f.type()
             3
 
-        TEST:
+        TESTS:
 
         We test that types can be given by an int (see :trac:`10699`)::
 
@@ -593,7 +593,7 @@ def __init__(self, faces, face_contour=None):
             sage: P
             Patch: [[(0, 0, 0), 1]*, [(0, 0, 0), 2]*, [(0, 0, 0), 3]*]
 
-        TEST:
+        TESTS:
 
         We test that colors are not anymore mixed up between
         Patches (see :trac:`11255`)::
@@ -670,7 +670,7 @@ def __hash__(self):
             sage: hash(P)      #random
             -4839605361791007520
 
-        TEST:
+        TESTS:
 
         We test that two equal patches have the same hash (see :trac:`11255`)::
 
@@ -1484,7 +1484,7 @@ def __call__(self, patch, iterations=1):
             sage: E(P, iterations=4)
             Patch of 31 faces
 
-        TEST:
+        TESTS:
 
         We test that iterations=0 works (see :trac:`10699`)::
 
diff --git a/src/sage/combinat/finite_state_machine.py b/src/sage/combinat/finite_state_machine.py
index 1dd58caa480..b911a8bef23 100644
--- a/src/sage/combinat/finite_state_machine.py
+++ b/src/sage/combinat/finite_state_machine.py
@@ -4328,7 +4328,7 @@ def format_transition_label_reversed(self, word):
             \path[->] (v0) edge[loop above] node {$0\mid 3 2 1$} ();
             \end{tikzpicture}
 
-        TEST:
+        TESTS:
 
         Check that :trac:`16357` is fixed::
 
@@ -4409,7 +4409,7 @@ def default_format_transition_label(self, word):
                 \path[->] (v0) edge node[rotate=360.00, anchor=south] {$t$} (v1);
                 \end{tikzpicture}
 
-        TEST:
+        TESTS:
 
         Check that :trac:`16357` is fixed::
 
@@ -11034,7 +11034,7 @@ def language(self, max_length=None, **kwargs):
           ``max_length`` will be considered. If ``None``, then this
           iterates over all possible words without length restrictions.
 
-        - ``kwargs`` -- will be passed on to to the :class:`process
+        - ``kwargs`` -- will be passed on to the :class:`process
           iterator <FSMProcessIterator>`. See :meth:`process` for a
           description.
 
@@ -12309,7 +12309,7 @@ def language(self, max_length=None, **kwargs):
           considered. If ``None``, then this iterates over all
           possible words without length restrictions.
 
-        - ``kwargs`` -- will be passed on to to the :class:`process
+        - ``kwargs`` -- will be passed on to the :class:`process
           iterator <FSMProcessIterator>`. See :meth:`process` for a
           description.
 
diff --git a/src/sage/combinat/kazhdan_lusztig.py b/src/sage/combinat/kazhdan_lusztig.py
index 41e08fe759a..55164f5236d 100644
--- a/src/sage/combinat/kazhdan_lusztig.py
+++ b/src/sage/combinat/kazhdan_lusztig.py
@@ -8,22 +8,26 @@
 - Alan J.X. Guo (2014-03-18): ``R_tilde()`` method.
 
 """
+
 #*****************************************************************************
-#       Copyright (C) 2008 Daniel Bump <bump at match.stanford.edu>,
-#
-#  Distributed under the terms of the GNU General Public License (GPL)
+#       Copyright (C) 2008 Daniel Bump <bump at match.stanford.edu>
 #
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+
+from __future__ import absolute_import, print_function
 
 from sage.rings.polynomial.polynomial_element import is_Polynomial
 from sage.functions.other import floor
 from sage.misc.cachefunc import cached_method
-from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial_generic
+from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
 from sage.structure.sage_object import SageObject
 from sage.structure.unique_representation import UniqueRepresentation
-from sage.combinat.root_system.coxeter_group import CoxeterGroup
+
 
 class KazhdanLusztigPolynomial(UniqueRepresentation, SageObject):
     """
@@ -87,7 +91,7 @@ def __init__(self, W, q, trace=False):
         self._base_ring = q.parent()
         if is_Polynomial(q):
             self._base_ring_type = "polynomial"
-        elif isinstance(q, LaurentPolynomial_generic):
+        elif isinstance(q, LaurentPolynomial):
             self._base_ring_type = "laurent"
         else:
             self._base_ring_type = "unknown"
diff --git a/src/sage/combinat/matrices/hadamard_matrix.py b/src/sage/combinat/matrices/hadamard_matrix.py
index 321f4d57a1e..9938ab331d0 100644
--- a/src/sage/combinat/matrices/hadamard_matrix.py
+++ b/src/sage/combinat/matrices/hadamard_matrix.py
@@ -546,8 +546,8 @@ def regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e,existence=False
 
     Other hardcoded values::
 
-        sage: for n,e in [(36,1),(36,-1),(100,1),(100,-1),(196, 1)]:
-        ....:     print(regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e))
+        sage: for n,e in [(36,1),(36,-1),(100,1),(100,-1),(196, 1)]:  # long time
+        ....:     print(repr(regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)))
         36 x 36 dense matrix over Integer Ring
         36 x 36 dense matrix over Integer Ring
         100 x 100 dense matrix over Integer Ring
@@ -555,24 +555,24 @@ def regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e,existence=False
         196 x 196 dense matrix over Integer Ring
 
         sage: for n,e in [(324,1),(324,-1)]: # not tested - long time, tested in RSHCD_324
-        ....:     print(regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)) # not tested - long time
+        ....:     print(repr(regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)))
         324 x 324 dense matrix over Integer Ring
         324 x 324 dense matrix over Integer Ring
 
     From two close prime powers::
 
-        sage: print(regular_symmetric_hadamard_matrix_with_constant_diagonal(64,-1))
-        64 x 64 dense matrix over Integer Ring
+        sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(64,-1)
+        64 x 64 dense matrix over Integer Ring (use the '.str()' method to see the entries)
 
     From a prime power and a conference matrix::
 
-        sage: print(regular_symmetric_hadamard_matrix_with_constant_diagonal(676,1)) # long time
-        676 x 676 dense matrix over Integer Ring
+        sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(676,1)  # long time
+        676 x 676 dense matrix over Integer Ring (use the '.str()' method to see the entries)
 
     Recursive construction::
 
-        sage: print(regular_symmetric_hadamard_matrix_with_constant_diagonal(144,-1))
-        144 x 144 dense matrix over Integer Ring
+        sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(144,-1)
+        144 x 144 dense matrix over Integer Ring (use the '.str()' method to see the entries)
 
     REFERENCE:
 
@@ -715,10 +715,10 @@ def RSHCD_324(e):
     TESTS::
 
         sage: from sage.combinat.matrices.hadamard_matrix import RSHCD_324, is_hadamard_matrix
-        sage: for e in [1,-1]: # long time
-        ....:     M = RSHCD_324(e) # long time
-        ....:     print("{} {} {}".format(M==M.T,is_hadamard_matrix(M),all([M[i,i]==1 for i in range(324)]))) # long time
-        ....:     print(set(map(sum,M))) # long time
+        sage: for e in [1,-1]:  # long time
+        ....:     M = RSHCD_324(e)
+        ....:     print("{} {} {}".format(M==M.T,is_hadamard_matrix(M),all([M[i,i]==1 for i in range(324)])))
+        ....:     print(set(map(sum,M)))
         True True True
         set([18])
         True True True
@@ -1317,7 +1317,7 @@ def rshcd_from_prime_power_and_conference_matrix(n):
     Bigger examples, only provided by this construction ::
 
         sage: H = rshcd_from_prime_power_and_conference_matrix(27)  # long time
-        sage: H==H.T and is_hadamard_matrix(H)                      # long time
+        sage: H == H.T and is_hadamard_matrix(H)                    # long time
         True
         sage: H.diagonal()==[1]*676 and list(sum(H))==[26]*676      # long time
         True
diff --git a/src/sage/combinat/plane_partition.py b/src/sage/combinat/plane_partition.py
index f16f92e2ac1..8a72dd62b05 100644
--- a/src/sage/combinat/plane_partition.py
+++ b/src/sage/combinat/plane_partition.py
@@ -220,58 +220,81 @@ def _repr_diagram(self, show_box=False, use_unicode=False):
         EXAMPLES::
 
             sage: print(PlanePartition([[4,3,3,1],[2,1,1],[1,1]])._repr_diagram())
-                 / \
-                |\ /|
-                |\|/ \
-               / \|\ / \
-              |\ /|\|\ /|
-             / \|/ \|\|/|
-            |\ / \ / \|/ \
-             \|\ /|\ /|\ /|
-               \|/ \|/ \|/
+                    __
+                   /\_\
+                __/\/_/
+             __/\_\/\_\
+            /\_\/_/\/\_\
+            \/\_\_\/\/_/
+             \/_/\_\/_/
+                \/_/\_\
+                   \/_/
             sage: print(PlanePartition([[4,3,3,1],[2,1,1],[1,1]])._repr_diagram(True))
-                 / \
-               /|\ /|\
-             /|/|\|/ \|\
-            |/|/ \|\ / \|\
-            |/|\ /|\|\ /|\|
-            |/ \|/ \|\|/|\|
-            |\ / \ / \|/ \|
-             \|\ /|\ /|\ /|
-               \|/ \|/ \|/
-                 \ / \ /
-                   \ /
+                ______
+               /_/_/\_\
+              /_/_/\/_/\
+             /_/\_\/\_\/\
+            /\_\/_/\/\_\/\
+            \/\_\_\/\/_/\/
+             \/_/\_\/_/\/
+              \_\/_/\_\/
+               \_\_\/_/
         """
         x = self._max_x
         y = self._max_y
         z = self._max_z
-        drawing = [[" " for i in range(2*x+2*y+1)] for j in range(x+y+z)]
-        def add_topside(i,j,k):
-            drawing[z+0+i+j-k][2*x-1-2*i+2*j] = u"╱" if use_unicode else "/"
-            drawing[z+0+i+j-k][2*x+1-2*i+2*j] = u"╲" if use_unicode else "\\"
-            drawing[z+1+i+j-k][2*x-1-2*i+2*j] = u"╲" if use_unicode else "\\"
-            drawing[z+1+i+j-k][2*x+1-2*i+2*j] = u"╱" if use_unicode else "/"
-        def add_rightside(i,j,k):
-            drawing[z+0+i+j-k][2*x-2-2*i+2*j] = u"│" if use_unicode else "|"
-            drawing[z+0+i+j-k][2*x-1-2*i+2*j] = u"╱" if use_unicode else "/"
-            drawing[z-1+i+j-k][2*x-1-2*i+2*j] = u"╱" if use_unicode else "/"
-            drawing[z-1+i+j-k][2*x-0-2*i+2*j] = u"│" if use_unicode else "|"
-        def add_leftside(i,j,k):
-            drawing[z-1+i+j-k][2*x-0-2*i+2*j] = u"│" if use_unicode else "|"
-            drawing[z-1+i+j-k][2*x+1-2*i+2*j] = u"╲" if use_unicode else "\\"
-            drawing[z+0+i+j-k][2*x+2-2*i+2*j] = u"│" if use_unicode else "|"
-            drawing[z+0+i+j-k][2*x+1-2*i+2*j] = u"╲" if use_unicode else "\\"
+
+        drawing = [[" " for i in range(2 * x + y + z)]
+                   for j in range(y + z + 1)]
+
+        hori = u"_" if use_unicode else "_"
+        down = u"╲" if use_unicode else "\\"
+        up = u"╱" if use_unicode else "/"
+
+        def superpose(l, c, letter):
+            # add the given letter at line l and column c
+            exist = drawing[l][c]
+            if exist == " " or exist == "_":
+                drawing[l][c] = letter
+
+        def add_topside(i, j, k):
+            X = z + j - k
+            Y = 2 * x - 2 * i + j + k
+            superpose(X, Y-2, hori)
+            superpose(X, Y-1, hori)
+            superpose(X + 1, Y-2, down)
+            superpose(X + 1, Y-1, hori)
+            superpose(X + 1, Y, down)
+
+        def add_rightside(i, j, k):
+            X = z + j - k
+            Y = 2 * x - 2 * i + j + k
+            superpose(X - 1, Y - 1, hori)
+            superpose(X - 1, Y, hori)
+            superpose(X, Y - 2, up)
+            superpose(X, Y - 1, hori)
+            superpose(X, Y, up)
+
+        def add_leftside(i, j, k):
+            X = z + j - k
+            Y = 2 * x - 2 * i + j + k
+            superpose(X, Y, up)
+            superpose(X, Y + 1, down)
+            superpose(X + 1, Y + 1, up)
+            superpose(X + 1, Y, down)
 
         tab = self.z_tableau()
         for r in range(len(tab)):
             for c in range(len(tab[r])):
                 if tab[r][c] > 0 or show_box:
                     add_topside(r, c, tab[r][c])
+
         tab = self.y_tableau()
         for r in range(len(tab)):
             for c in range(len(tab[r])):
                 if self.y_tableau()[r][c] > 0 or show_box:
                     add_rightside(c, tab[r][c], r)
+
         tab = self.x_tableau()
         for r in range(len(tab)):
             for c in range(len(tab[r])):
@@ -298,16 +321,17 @@ def _ascii_art_(self):
 
         EXAMPLES::
 
-            sage: print(PlanePartition([[4,3,3,1],[2,1,1],[1,1]])._ascii_art_())
-                 / \
-                |\ /|
-                |\|/ \
-               / \|\ / \
-              |\ /|\|\ /|
-             / \|/ \|\|/|
-            |\ / \ / \|/ \
-             \|\ /|\ /|\ /|
-               \|/ \|/ \|/
+            sage: PP = PlanePartition([[4,3,3,1],[2,1,1],[1,1]])
+            sage: ascii_art(PP)
+                    __
+                   /\_\
+                __/\/_/
+             __/\_\/\_\
+            /\_\/_/\/\_\
+            \/\_\_\/\/_/
+             \/_/\_\/_/
+                \/_/\_\
+                   \/_/
         """
         from sage.typeset.ascii_art import AsciiArt
         return AsciiArt(self._repr_diagram().splitlines(), baseline=0)
@@ -320,15 +344,15 @@ def _unicode_art_(self):
 
             sage: PP = PlanePartition([[4,3,3,1],[2,1,1],[1,1]])
             sage: unicode_art(PP)
-                 ╱ ╲
-                │╲ ╱│
-                │╲│╱ ╲
-               ╱ ╲│╲ ╱ ╲
-              │╲ ╱│╲│╲ ╱│
-             ╱ ╲│╱ ╲│╲│╱│
-            │╲ ╱ ╲ ╱ ╲│╱ ╲
-             ╲│╲ ╱│╲ ╱│╲ ╱│
-               ╲│╱ ╲│╱ ╲│╱
+                    __
+                   ╱╲_╲
+                __╱╲╱_╱
+             __╱╲_╲╱╲_╲
+            ╱╲_╲╱_╱╲╱╲_╲
+            ╲╱╲_╲_╲╱╲╱_╱
+             ╲╱_╱╲_╲╱_╱
+                ╲╱_╱╲_╲
+                   ╲╱_╱
         """
         from sage.typeset.unicode_art import UnicodeArt
         return UnicodeArt(self._repr_diagram(use_unicode=True).splitlines(), baseline=0)
@@ -349,27 +373,25 @@ def pp(self, show_box=False):
         EXAMPLES::
 
             sage: PlanePartition([[4,3,3,1],[2,1,1],[1,1]]).pp()
-                 / \
-                |\ /|
-                |\|/ \
-               / \|\ / \
-              |\ /|\|\ /|
-             / \|/ \|\|/|
-            |\ / \ / \|/ \
-             \|\ /|\ /|\ /|
-               \|/ \|/ \|/
+                    __
+                   /\_\
+                __/\/_/
+             __/\_\/\_\
+            /\_\/_/\/\_\
+            \/\_\_\/\/_/
+             \/_/\_\/_/
+                \/_/\_\
+                   \/_/
             sage: PlanePartition([[4,3,3,1],[2,1,1],[1,1]]).pp(True)
-                 / \
-               /|\ /|\
-             /|/|\|/ \|\
-            |/|/ \|\ / \|\
-            |/|\ /|\|\ /|\|
-            |/ \|/ \|\|/|\|
-            |\ / \ / \|/ \|
-             \|\ /|\ /|\ /|
-               \|/ \|/ \|/
-                 \ / \ /
-                   \ /
+                ______
+               /_/_/\_\
+              /_/_/\/_/\
+             /_/\_\/\_\/\
+            /\_\/_/\/\_\/\
+            \/\_\_\/\/_/\/
+             \/_/\_\/_/\/
+              \_\/_/\_\/
+               \_\_\/_/
         """
         print(self._repr_diagram(show_box))
 
@@ -836,4 +858,3 @@ def leq(thing1, thing2):
         return self.element_class(self, Z, check=False)
 
     Element = PlanePartition
-
diff --git a/src/sage/combinat/posets/poset_examples.py b/src/sage/combinat/posets/poset_examples.py
index 89d53423ca8..8194158b4de 100644
--- a/src/sage/combinat/posets/poset_examples.py
+++ b/src/sage/combinat/posets/poset_examples.py
@@ -1630,7 +1630,7 @@ def _random_stone_lattice(n):
     ALGORITHM:
 
     Randomly split `n` to some factors. For every factor `p` generate
-    a random distributive lattice on `p-1` elements and add a new new bottom
+    a random distributive lattice on `p-1` elements and add a new bottom
     element to it. Compute the cartesian product of those lattices.
     """
     from sage.arith.misc import factor
diff --git a/src/sage/combinat/posets/posets.py b/src/sage/combinat/posets/posets.py
index 6a01feb5345..ad4d766e149 100644
--- a/src/sage/combinat/posets/posets.py
+++ b/src/sage/combinat/posets/posets.py
@@ -1462,8 +1462,6 @@ def linear_extension(self, linear_extension=None, check=True):
           whether to check that ``linear_extension`` is indeed a
           linear extension of ``self``.
 
-        .. SEEALSO:: :meth:`is_linear_extension`, :meth:`linear_extensions`
-
         EXAMPLES::
 
             sage: P = Poset((divisors(15), attrcall("divides")), facade=True)
@@ -1496,6 +1494,8 @@ def linear_extension(self, linear_extension=None, check=True):
             sage: P.linear_extension([1,3,15,5], check=False)
             [1, 3, 15, 5]
 
+        .. SEEALSO:: :meth:`is_linear_extension`, :meth:`linear_extensions`
+
         .. TODO::
 
             - Is it acceptable to have those two features for a single method?
@@ -1530,8 +1530,6 @@ def linear_extensions(self, facade=False):
                 ...
                 TypeError: Cannot convert list to sage.structure.element.Element
 
-        .. SEEALSO:: :meth:`linear_extension`, :meth:`is_linear_extension`
-
         EXAMPLES::
 
             sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
@@ -1576,6 +1574,8 @@ def linear_extensions(self, facade=False):
                 sage: type(L[0])
                 <... 'list'>
 
+        .. SEEALSO:: :meth:`linear_extension`, :meth:`is_linear_extension`
+
         TESTS::
 
             sage: D = Poset({ 0:[1,2], 1:[3], 2:[3,4] })
@@ -1594,8 +1594,6 @@ def is_linear_extension(self, l):
 
         - ``l`` -- a list (or iterable) containing all of the elements of ``self`` exactly once
 
-        .. SEEALSO:: :meth:`linear_extension`, :meth:`linear_extensions`
-
         EXAMPLES::
 
             sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
@@ -1624,6 +1622,8 @@ def is_linear_extension(self, l):
             This is used and systematically tested in
             :class:`~sage.combinat.posets.linear_extensions.LinearExtensionsOfPosets`
 
+        .. SEEALSO:: :meth:`linear_extension`, :meth:`linear_extensions`
+
         TESTS:
 
         Check that :trac:`15313` is fixed::
@@ -1905,11 +1905,6 @@ def level_sets(self):
 
         Every level is an antichain of the poset.
 
-        .. SEEALSO::
-
-            :meth:`dilworth_decomposition` to return elements grouped
-            to chains.
-
         EXAMPLES::
 
             sage: P = Poset({0:[1,2],1:[3],2:[3],3:[]})
@@ -1919,6 +1914,11 @@ def level_sets(self):
             sage: Q = Poset({0:[1,2], 1:[3], 2:[4], 3:[4]})
             sage: Q.level_sets()
             [[0], [1, 2], [3], [4]]
+
+        .. SEEALSO::
+
+            :meth:`dilworth_decomposition` to return elements grouped
+            to chains.
         """
         return [[self._vertex_to_element(_) for _ in level] for level in
                 self._hasse_diagram.level_sets()]
@@ -2003,10 +2003,6 @@ def relations(self):
         A list of pairs (each pair is a list), where the first element
         of the pair is less than or equal to the second element.
 
-        .. SEEALSO::
-
-            :meth:`relations_number`, :meth:`relations_iterator`
-
         EXAMPLES::
 
             sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
@@ -2014,6 +2010,10 @@ def relations(self):
             [[1, 1], [1, 2], [1, 3], [1, 4], [0, 0], [0, 2], [0, 3],
             [0, 4], [2, 2], [2, 3], [2, 4], [3, 3], [3, 4], [4, 4]]
 
+        .. SEEALSO::
+
+            :meth:`relations_number`, :meth:`relations_iterator`
+
         TESTS::
 
             sage: P = Poset()  # Test empty poset
@@ -2107,10 +2107,6 @@ def relations_iterator(self, strict=False):
         A generator that produces pairs (each pair is a list), where the
         first element of the pair is less than or equal to the second element.
 
-        .. SEEALSO::
-
-            :meth:`relations_number`, :meth:`relations`.
-
         EXAMPLES::
 
             sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
@@ -2124,6 +2120,10 @@ def relations_iterator(self, strict=False):
             sage: list(P.relations_iterator(strict=True))
             [[0, 1], [0, 2], [0, 4], [0, 3], [1, 4], [2, 3], [2, 4], [3, 4]]
 
+        .. SEEALSO::
+
+            :meth:`relations_number`, :meth:`relations`.
+
         AUTHOR:
 
         - Rob Beezer (2011-05-04)
@@ -2152,10 +2152,6 @@ def relations_number(self):
         Relations are also often called intervals. The number of
         intervals is the dimension of the incidence algebra.
 
-        .. SEEALSO::
-
-            :meth:`relations_iterator`, :meth:`relations`
-
         EXAMPLES::
 
             sage: P = Posets.PentagonPoset()
@@ -2166,6 +2162,10 @@ def relations_number(self):
             sage: TamariLattice(4).relations_number()
             68
 
+        .. SEEALSO::
+
+            :meth:`relations_iterator`, :meth:`relations`
+
         TESTS::
 
             sage: Poset().relations_number()
@@ -2181,7 +2181,8 @@ def is_incomparable_chain_free(self, m, n=None):
         Return ``True`` if the poset is `(m+n)`-free, and ``False`` otherwise.
 
         A poset is `(m+n)`-free if there is no incomparable chains of
-        lengths `m` and `n`. Three cases have special name:
+        lengths `m` and `n`. Three cases have special name
+        (see [EnumComb1]_, exercise 3.15):
 
         - ''interval order'' is `(2+2)`-free
         - ''semiorder'' (or ''unit interval order'') is `(1+3)`-free and
@@ -2266,14 +2267,6 @@ def is_incomparable_chain_free(self, m, n=None):
         AUTHOR:
 
         - Eric Rowland (2013-05-28)
-
-        REFERENCES:
-
-        .. [EnumComb1] Richard P. Stanley,
-           *Enumerative Combinatorics, volume 1*,
-           Second Edition,
-           Cambridge University Press (2011).
-           http://math.mit.edu/~rstan/ec/ec1/
         """
         if n is None:
             try:
@@ -2498,7 +2491,11 @@ def has_bottom(self):
             sage: Q.has_bottom()
             True
 
-        .. SEEALSO:: :meth:`bottom`, :meth:`has_top`, :meth:`is_bounded`
+        .. SEEALSO::
+
+            - Dual Property: :meth:`has_top`
+            - Stronger properties: :meth:`is_bounded`
+            - Other: :meth:`bottom`
 
         TESTS::
 
@@ -2550,7 +2547,11 @@ def has_top(self):
             sage: Q.has_top()
             True
 
-        .. SEEALSO:: :meth:`top`, :meth:`has_bottom`, :meth:`is_bounded`
+        .. SEEALSO::
+
+            - Dual Property: :meth:`has_bottom`
+            - Stronger properties: :meth:`is_bounded`
+            - Other: :meth:`top`
 
         TESTS::
 
@@ -2658,7 +2659,10 @@ def is_bounded(self):
             sage: Q.is_bounded()
             True
 
-        .. SEEALSO:: :meth:`has_bottom`, :meth:`has_top`
+        .. SEEALSO::
+
+            - Weaker properties: :meth:`has_bottom`, :meth:`has_top`
+            - Other: :meth:`with_bounds`, :meth:`without_bounds`
 
         TESTS::
 
@@ -2893,13 +2897,6 @@ def is_EL_labelling(self, f, return_raising_chains=False):
              ((0, 0), (1, 1)): [0, 1],
              ((0, 1), (1, 1)): [0],
              ((1, 0), (1, 1)): [1]}
-
-        REFERENCES:
-
-        .. [Bj1980] Anders Björner,
-           *Shellable and Cohen-Macaulay partially ordered sets*,
-           Trans. Amer. Math. Soc. 260 (1980), 159-183,
-           :doi:`10.1090/S0002-9947-1980-0570784-2`
         """
         label_dict = { (a,b):f(a,b) for a,b in self.cover_relations_iterator() }
         if return_raising_chains:
@@ -3001,14 +2998,6 @@ def dimension(self, certificate=False):
             0
             sage: Poset().dimension(certificate=True)
             []
-
-        REFERENCES:
-
-        .. [FT00] Stefan Felsner, William T. Trotter,
-           Dimension, Graph and Hypergraph Coloring,
-           Order,
-           2000, Volume 17, Issue 2, pp 167-177,
-           http://link.springer.com/article/10.1023%2FA%3A1006429830221
         """
         if self.cardinality() == 0:
             return [] if certificate else 0
@@ -3127,13 +3116,9 @@ def jump_number(self, certificate=False):
             sage: N.jump_number(certificate=True)
             (1, [1, 4, 2, 3])
 
-        REFERENCES:
+        ALGORITHM:
 
-        .. [BIANCO] L. Bianco, P. Dell‘Olmo, S. Giordani
-           An Optimal Algorithm to Find the Jump Number of Partially Ordered Sets
-           Computational Optimization and Applications,
-           1997, Volume 8, Issue 2, pp 197--210,
-           :doi:`10.1023/A:1008625405476`
+        See [BIANCO]_
 
         TESTS::
 
@@ -3422,14 +3407,14 @@ def cardinality(self):
         """
         Return the number of elements in the poset.
 
-        .. SEEALSO::
-
-            :meth:`degree_polynomial` for a more refined invariant
-
         EXAMPLES::
 
             sage: Poset([[1,2,3],[4],[4],[4],[]]).cardinality()
             5
+
+        .. SEEALSO::
+
+            :meth:`degree_polynomial` for a more refined invariant
         """
         return Integer(self._hasse_diagram.order())
 
@@ -3571,10 +3556,6 @@ def coxeter_transformation(self):
         modules. This matrix is usually called the Coxeter
         transformation.
 
-        .. SEEALSO::
-
-            :meth:`coxeter_polynomial`
-
         EXAMPLES::
 
             sage: Posets.PentagonPoset().coxeter_transformation()
@@ -3584,6 +3565,10 @@ def coxeter_transformation(self):
             [-1  1  1  0 -1]
             [-1  1  0  1 -1]
 
+        .. SEEALSO::
+
+            :meth:`coxeter_polynomial`
+
         TESTS::
 
             sage: M = posets.PentagonPoset().coxeter_transformation()
@@ -3604,10 +3589,6 @@ def coxeter_polynomial(self):
         transformation. This polynomial only depends on the derived
         category of modules on the poset.
 
-        .. SEEALSO::
-
-            :meth:`coxeter_transformation`
-
         EXAMPLES::
 
             sage: P = posets.PentagonPoset()
@@ -3616,7 +3597,11 @@ def coxeter_polynomial(self):
 
             sage: p = posets.SymmetricGroupWeakOrderPoset(3)
             sage: p.coxeter_polynomial()
-            x^6 + x^5 - x^3 + x + 1
+            x^6 + x^5 - x^3 + x + 1 
+
+        .. SEEALSO::
+
+            :meth:`coxeter_transformation`
         """
         return self._hasse_diagram.coxeter_transformation().charpoly()
 
@@ -3653,7 +3638,10 @@ def is_meet_semilattice(self, certificate=False):
             sage: V.is_meet_semilattice(certificate=True)
             (False, ((2, 2, 1), (3, 1, 1)))
 
-        .. SEEALSO:: :meth:`is_join_semilattice`, :meth:`~sage.categories.finite_posets.FinitePosets.ParentMethods.is_lattice`
+        .. SEEALSO::
+
+            - Dual property: :meth:`is_join_semilattice`
+            - Stronger properties: :meth:`~sage.categories.finite_posets.FinitePosets.ParentMethods.is_lattice`
 
         TESTS::
 
@@ -3721,7 +3709,10 @@ def is_join_semilattice(self, certificate=False):
             sage: P.is_join_semilattice(certificate=True)
             (False, (2, 1))
 
-        .. SEEALSO:: :meth:`is_meet_semilattice`, :meth:`~sage.categories.finite_posets.FinitePosets.ParentMethods.is_lattice`
+        .. SEEALSO::
+
+            - Dual property: :meth:`is_meet_semilattice`
+            - Stronger properties: :meth:`~sage.categories.finite_posets.FinitePosets.ParentMethods.is_lattice`
 
         TESTS::
 
@@ -3788,10 +3779,6 @@ def isomorphic_subposets_iterator(self, other):
 
         - ``other`` -- a finite poset
 
-        .. SEEALSO::
-
-            :meth:`sage.combinat.posets.lattices.FiniteLatticePoset.isomorphic_sublattices_iterator`.
-
         EXAMPLES::
 
             sage: D = Poset({1:[2,3], 2:[4], 3:[4]})
@@ -3812,6 +3799,9 @@ def isomorphic_subposets_iterator(self, other):
             function does not eat memory like
             :meth:`isomorphic_subposets` does.
 
+        .. SEEALSO::
+
+            :meth:`sage.combinat.posets.lattices.FiniteLatticePoset.isomorphic_sublattices_iterator`.
         """
         if not hasattr(other, 'hasse_diagram'):
             raise TypeError("'other' is not a finite poset")
@@ -4005,10 +3995,6 @@ def dilworth_decomposition(self):
 
         See :wikipedia:`Dilworth's_theorem`.
 
-        .. SEEALSO::
-
-            :meth:`level_sets` to return elements grouped to antichains.
-
         ALGORITHM:
 
         We build a bipartite graph in which a vertex `v` of the poset is
@@ -4031,6 +4017,11 @@ def dilworth_decomposition(self):
             sage: p.dilworth_decomposition()  # random
             [[7, 6, 4], [11, 3], [12, 8, 0], [13, 9, 1], [14, 10, 2], [15, 5]]
 
+
+        .. SEEALSO::
+
+            :meth:`level_sets` to return elements grouped to antichains.
+
         TESTS::
 
             sage: p = posets.IntegerCompositions(5)
@@ -4502,11 +4493,6 @@ def ordinal_sum(self, other, labels='pairs'):
           result. If set to 'integers', the elements of the result
           will be relabeled with consecutive integers.
 
-        .. SEEALSO::
-
-            :meth:`ordinal_summands`, :meth:`disjoint_union`,
-            :meth:`sage.combinat.posets.lattices.FiniteLatticePoset.vertical_composition`
-
         EXAMPLES::
 
             sage: P1 = Poset( ([1, 2, 3, 4], [[1, 2], [1, 3], [1, 4]]) )
@@ -4540,6 +4526,11 @@ def ordinal_sum(self, other, labels='pairs'):
             sage: L.ordinal_sum(L)
             Finite lattice containing 4 elements
 
+        .. SEEALSO::
+
+            :meth:`ordinal_summands`, :meth:`disjoint_union`,
+            :meth:`sage.combinat.posets.lattices.FiniteLatticePoset.vertical_composition`
+
         TESTS::
 
             sage: N5 = Posets.PentagonPoset()
@@ -4736,10 +4727,6 @@ def with_bounds(self, labels=('bottom', 'top')):
           ``'top'``. Either of them can be ``None``, and then a new
           bottom or top element will not be added.
 
-        .. SEEALSO::
-
-            :meth:`without_bounds` for the reverse operation
-
         EXAMPLES::
 
             sage: V = Poset({0: [1, 2]})
@@ -4759,6 +4746,10 @@ def with_bounds(self, labels=('bottom', 'top')):
             sage: P.with_bounds()
             Finite lattice containing 7 elements
 
+        .. SEEALSO::
+
+            :meth:`without_bounds` for the reverse operation
+
         TESTS::
 
             sage: P = Poset().with_bounds()
@@ -4873,10 +4864,6 @@ def without_bounds(self):
         If there is either no top or no bottom element, this
         raises a ``TypeError``.
 
-        .. SEEALSO::
-
-            :meth:`with_bounds` for the reverse operation
-
         EXAMPLES::
 
             sage: P = posets.PentagonPoset()
@@ -4891,6 +4878,10 @@ def without_bounds(self):
             sage: Q.cover_relations()
             []
 
+        .. SEEALSO::
+
+            :meth:`with_bounds` for the reverse operation
+
         TESTS::
 
             sage: P = Poset({1:[2],3:[2,4]})
@@ -5075,11 +5066,6 @@ def canonical_label(self, algorithm=None):
         - ``algorithm`` -- string (optional); a parameter forwarded
           to underlying graph function to select the algorithm to use
 
-        .. SEEALSO::
-
-            - Canonical labeling of directed graphs:
-              :meth:`~sage.graphs.generic_graph.GenericGraph.canonical_label()`
-
         EXAMPLES::
 
             sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
@@ -5101,6 +5087,11 @@ def canonical_label(self, algorithm=None):
             sage: ML.canonical_label()
             Finite meet-semilattice containing 3 elements
 
+        .. SEEALSO::
+
+            - Canonical labeling of directed graphs:
+              :meth:`~sage.graphs.generic_graph.GenericGraph.canonical_label()`
+
         TESTS::
 
             sage: P = Poset(digraphs.Path(10), linear_extension=True)
@@ -5194,9 +5185,8 @@ def graphviz_string(self,graph_string="graph",edge_string="--"):
         Returns a representation in the DOT language, ready to render in
         graphviz.
 
-        REFERENCES:
-
-        - http://www.graphviz.org/doc/info/lang.html
+        See http://www.graphviz.org/doc/info/lang.html for more information
+        about graphviz.
 
         EXAMPLES::
 
@@ -5335,14 +5325,6 @@ def random_order_ideal(self, direction='down'):
             sage: a = P.random_order_ideal('down')
             sage: P.is_order_ideal(a)
             True
-
-        REFERENCES:
-
-        .. [Propp1997] James Propp,
-           *Generating Random Elements of Finite Distributive Lattices*,
-           Electron. J. Combin. 4 (1997), no. 2, The Wilf Festschrift volume,
-           Research Paper 15.
-           http://www.combinatorics.org/ojs/index.php/eljc/article/view/v4i2r15
         """
         from sage.misc.randstate import current_randstate
         from sage.misc.randstate import seed
@@ -5549,8 +5531,6 @@ def comparability_graph(self):
 
         See :wikipedia:`Comparability_graph`
 
-        .. SEEALSO:: :meth:`incomparability_graph`, :mod:`sage.graphs.comparability`
-
         EXAMPLES::
 
             sage: Y = Poset({1: [2], 2: [3, 4]})
@@ -5561,6 +5541,8 @@ def comparability_graph(self):
             sage: g.has_edge(1, 3)
             True
 
+        .. SEEALSO:: :meth:`incomparability_graph`, :mod:`sage.graphs.comparability`
+
         TESTS::
 
             sage: Poset().comparability_graph()
@@ -5587,8 +5569,6 @@ def incomparability_graph(self):
         and there is an edge between vertices if they are not
         comparable in the poset.
 
-        .. SEEALSO:: :meth:`comparability_graph`
-
         EXAMPLES::
 
             sage: Y = Poset({1: [2], 2: [3, 4]})
@@ -5599,6 +5579,8 @@ def incomparability_graph(self):
             sage: g.has_edge(1, 3)
             False
 
+        .. SEEALSO:: :meth:`comparability_graph`
+
         TESTS::
 
             sage: Poset().incomparability_graph()
@@ -5816,11 +5798,6 @@ def order_polytope(self):
             False
             sage: Q.contains((0,1,1))
             True
-
-        REFERENCES:
-
-        .. [St1986] Richard Stanley. *Two poset polytopes*,
-           Discrete Comput. Geom. (1986), :doi:`10.1007/BF02187680`
         """
         from sage.geometry.polyhedron.constructor import Polyhedron
         ineqs = [[0] + [ZZ(j==v) - ZZ(j==u) for j in self]
@@ -5940,11 +5917,6 @@ def f_polynomial(self):
         The coefficient of `q^i` is the number of chains of
         `i+1` elements containing both bounds of the poset.
 
-        .. SEEALSO::
-
-            :meth:`is_bounded`, :meth:`h_polynomial`, :meth:`order_complex`,
-            :meth:`sage.homology.cell_complex.GenericCellComplex.f_vector`
-
         .. note::
 
             This is slightly different from the ``fPolynomial``
@@ -5960,6 +5932,11 @@ def f_polynomial(self):
             sage: P.f_polynomial()
             q^4 + 4*q^3 + 5*q^2 + q
 
+        .. SEEALSO::
+
+            :meth:`is_bounded`, :meth:`h_polynomial`, :meth:`order_complex`,
+            :meth:`sage.homology.cell_complex.GenericCellComplex.f_vector`
+
         TESTS::
 
             sage: P = Poset({2: []})
@@ -5995,11 +5972,6 @@ def h_polynomial(self):
 
         See :wikipedia:`h-vector`.
 
-        .. SEEALSO::
-
-            :meth:`is_bounded`, :meth:`f_polynomial`, :meth:`order_complex`,
-            :meth:`sage.homology.simplicial_complex.SimplicialComplex.h_vector`
-
         .. WARNING::
 
             This is slightly different from the ``hPolynomial``
@@ -6016,6 +5988,11 @@ def h_polynomial(self):
             sage: P = Poset({1: []})
             sage: P.h_polynomial()
             1
+
+        .. SEEALSO::
+
+            :meth:`is_bounded`, :meth:`f_polynomial`, :meth:`order_complex`,
+            :meth:`sage.homology.simplicial_complex.SimplicialComplex.h_vector`
         """
         q = polygen(ZZ, 'q')
         hasse = self._hasse_diagram
@@ -6061,8 +6038,6 @@ def flag_f_polynomial(self):
 
         See :wikipedia:`h-vector`.
 
-        .. SEEALSO:: :meth:`is_bounded`, :meth:`flag_h_polynomial`
-
         EXAMPLES::
 
             sage: P = Posets.DiamondPoset(5)
@@ -6080,6 +6055,8 @@ def flag_f_polynomial(self):
             sage: P.flag_f_polynomial()
             3*x1*x2*x3 + 3*x1*x3 + x2*x3 + x3
 
+        .. SEEALSO:: :meth:`is_bounded`, :meth:`flag_h_polynomial`
+
         TESTS::
 
             sage: P = Poset({2: [3]})
@@ -6132,8 +6109,6 @@ def flag_h_polynomial(self):
 
         See :wikipedia:`h-vector`.
 
-        .. SEEALSO:: :meth:`is_bounded`, :meth:`flag_f_polynomial`
-
         EXAMPLES::
 
             sage: P = Posets.DiamondPoset(5)
@@ -6155,6 +6130,8 @@ def flag_h_polynomial(self):
             sage: P.flag_h_polynomial()
             x3
 
+        .. SEEALSO:: :meth:`is_bounded`, :meth:`flag_f_polynomial`
+
         TESTS::
 
             sage: P = Poset({2: [3]})
@@ -6297,8 +6274,6 @@ def order_polynomial(self):
         See sections 3.12 and 3.15 of [EnumComb1]_, and also
         [St1986]_.
 
-        .. SEEALSO:: :meth:`order_polytope`
-
         EXAMPLES::
 
             sage: P = Posets.AntichainPoset(3)
@@ -6310,6 +6285,8 @@ def order_polynomial(self):
             1/6*q^3 + 1/2*q^2 + 1/3*q
             sage: [f(i) for i in range(4)]
             [0, 1, 4, 10]
+
+        .. SEEALSO:: :meth:`order_polytope`
         """
         return self.order_ideals_lattice(as_ideals=False).zeta_polynomial()
 
@@ -6330,10 +6307,6 @@ def degree_polynomial(self):
         product of posets, it is useful to help see if the poset can
         be isomorphic to a Cartesian product.
 
-        .. SEEALSO::
-
-            :meth:`cardinality` for the value at `(x, y) = (1, 1)`
-
         EXAMPLES::
 
             sage: P = posets.PentagonPoset()
@@ -6343,6 +6316,10 @@ def degree_polynomial(self):
             sage: P = posets.BooleanLattice(4)
             sage: P.degree_polynomial().factor()
             (x + y)^4
+
+        .. SEEALSO::
+
+            :meth:`cardinality` for the value at `(x, y) = (1, 1)`
         """
         return self._hasse_diagram.degree_polynomial()
 
@@ -6450,21 +6427,6 @@ def evacuation(self):
         (see :meth:`~sage.combinat.posets.linear_extensions.LinearExtensionOfPoset.evacuation`),
         and relabeling ``self`` accordingly. For more details see [Stan2009]_.
 
-        .. SEEALSO::
-
-            - :meth:`linear_extension`
-            - :meth:`with_linear_extension` and the ``linear_extension`` option of :func:`Poset`
-            - :meth:`~sage.combinat.posets.linear_extensions.LinearExtensionOfPoset.evacuation`
-            - :meth:`promotion`
-
-        REFERENCES:
-
-        .. [Stan2009] Richard Stanley,
-           *Promotion and evacuation*,
-           Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of
-           Anders Björner,
-           Research Paper 9, 24 pp.
-
         EXAMPLES::
 
             sage: P = Poset(([1,2], [[1,2]]), linear_extension=True, facade=False)
@@ -6503,6 +6465,13 @@ def evacuation(self):
             sage: Q.cover_relations()
             [[1, 3], [1, 5], [3, 15], [5, 15]]
 
+        .. SEEALSO::
+
+            - :meth:`linear_extension`
+            - :meth:`with_linear_extension` and the ``linear_extension`` option of :func:`Poset`
+            - :meth:`~sage.combinat.posets.linear_extensions.LinearExtensionOfPoset.evacuation`
+            - :meth:`promotion`
+
         AUTHOR:
 
         - Anne Schilling (2012-02-18)
@@ -6783,13 +6752,6 @@ def frank_network(self):
                  (2, 0)                           t
                   a[e]                           a(e)
 
-        REFERENCES:
-
-        .. [BF1999] Thomas Britz, Sergey Fomin,
-           *Finite posets and Ferrers shapes*,
-           Advances in Mathematics 158, pp. 86-127 (2001),
-           :arxiv:`math/9912126` (the arXiv version has fewer errors).
-
         EXAMPLES::
 
             sage: ps = [[16,12,14,-13],[[12,14],[14,-13],[12,16],[16,-13]]]
@@ -7006,12 +6968,6 @@ def p_partition_enumerator(self, tup, R, weights=None, check=False):
             M[3, 5, 7] + M[3, 12] + M[5, 3, 7] + M[8, 7]
             sage: FP = P.p_partition_enumerator(('a','b','c'), QQ, weights={'a': 3, 'b': 5, 'c': 7}, check=True); FP
             M[3, 5, 7] + M[3, 12] + M[5, 3, 7] + M[5, 10] + M[8, 7] + M[15]
-
-        REFERENCES:
-
-        .. [Grinb2016a] Darij Grinberg,
-           *Double posets and the antipode of QSym*,
-           :arxiv:`1509.08355v2`.
         """
         if check:
             if sorted(self.list()) != sorted(tup):
@@ -7062,12 +7018,6 @@ def cuts(self):
         .. SEEALSO::
 
             :meth:`completion_by_cuts`
-
-        REFERENCES:
-
-        .. [JRJ94] Jourdan, Guy-Vincent; Rampon, Jean-Xavier; Jard, Claude
-           (1994), "Computing on-line the lattice of maximal antichains
-           of posets", Order 11 (3) p. 197-210, :doi:`10.1007/BF02115811`
         """
         from sage.graphs.graph import Graph
         from sage.graphs.independent_sets import IndependentSets
@@ -7090,10 +7040,6 @@ def completion_by_cuts(self):
 
         - a finite lattice
 
-        .. SEEALSO::
-
-            :meth:`~sage.categories.finite_lattice_posets.FiniteLatticePosets.ParentMethods.irreducibles_poset`
-
         EXAMPLES::
 
             sage: P = Posets.PentagonPoset()
@@ -7112,7 +7058,8 @@ def completion_by_cuts(self):
 
         .. SEEALSO::
 
-            :meth:`cuts`
+            :meth:`cuts`,
+            :meth:`~sage.categories.finite_lattice_posets.FiniteLatticePosets.ParentMethods.irreducibles_poset`
 
         TESTS::
 
@@ -7148,10 +7095,6 @@ def _kl_poly(self, x=None, y=None, canonical_labels=None):
         r"""
         Cached Kazhdan-Lusztig polynomial of ``self`` for generic `q`.
 
-        .. SEEALSO::
-
-            :meth:`kazhdan_lusztig_polynomial`
-
         EXAMPLES::
 
             sage: L = posets.SymmetricGroupWeakOrderPoset(4)
@@ -7162,6 +7105,10 @@ def _kl_poly(self, x=None, y=None, canonical_labels=None):
             sage: L._kl_poly(x, y)
             -q + 1
 
+        .. SEEALSO::
+
+            :meth:`kazhdan_lusztig_polynomial`
+
         AUTHORS:
 
         - Travis Scrimshaw (27-12-2014)
@@ -7259,12 +7206,6 @@ def kazhdan_lusztig_polynomial(self, x=None, y=None, q=None, canonical_labels=No
             sage: L.kazhdan_lusztig_polynomial(x, y, var('t'))
             -t + 1
 
-        REFERENCES:
-
-        .. [EPW14] Ben Elias, Nicholas Proudfoot, and Max Wakefield.
-           *The Kazhdan-Lusztig polynomial of a matroid*. 2014.
-           :arxiv:`1412.7408`.
-
         AUTHORS:
 
         - Travis Scrimshaw (27-12-2014)
diff --git a/src/sage/combinat/root_system/root_lattice_realization_algebras.py b/src/sage/combinat/root_system/root_lattice_realization_algebras.py
index 2c0d68d2848..f73281c410f 100644
--- a/src/sage/combinat/root_system/root_lattice_realization_algebras.py
+++ b/src/sage/combinat/root_system/root_lattice_realization_algebras.py
@@ -737,7 +737,7 @@ def demazure_lusztig_operators_on_classical(self, q, q1, q2, convention="antidom
             # In type BC dual we used q^2 and q elsewhere
             # Not sure this is the right thing to do or just a workaround ...
             # This probably makes up for the fact that, in type BC
-            # dual, the null null coroot is twice Sage's deltacheck
+            # dual, the null coroot is twice Sage's deltacheck
             # whereas the null root is delta. So we need to map delta
             # to q^2 in the q_projection.
             # Should this go in q_project instead?
diff --git a/src/sage/combinat/root_system/type_E.py b/src/sage/combinat/root_system/type_E.py
index 7cc835879c9..8df8417a8dd 100644
--- a/src/sage/combinat/root_system/type_E.py
+++ b/src/sage/combinat/root_system/type_E.py
@@ -159,7 +159,7 @@ def simple_root(self, i):
 
     def negative_roots(self):
         """
-        The negative negative roots.
+        The negative roots.
 
         EXAMPLES::
 
diff --git a/src/sage/combinat/skew_tableau.py b/src/sage/combinat/skew_tableau.py
index 0d343f4d533..aed3661b148 100644
--- a/src/sage/combinat/skew_tableau.py
+++ b/src/sage/combinat/skew_tableau.py
@@ -1579,7 +1579,7 @@ def is_k_tableau(self, k):
 
 def _label_skew(list_of_cells, sk):
     """
-    Return a filled-in standard standard skew tableau given an
+    Return a filled-in standard skew tableau given an
     ordered list ``list_of_cells`` of the coordinates to fill in
     (as pairs) and an empty shape ``sk``.
 
diff --git a/src/sage/combinat/subword.py b/src/sage/combinat/subword.py
index 0fcad680b57..6d941c4c414 100644
--- a/src/sage/combinat/subword.py
+++ b/src/sage/combinat/subword.py
@@ -523,7 +523,7 @@ def smallest_positions(word, subword, pos=0):
         sage: sage.combinat.subword.smallest_positions([1,3,3,5,4,5,3,5],[3,5,3],3)
         False
 
-    TEST:
+    TESTS:
 
     We check for :trac:`5534`::
 
diff --git a/src/sage/combinat/tutorial.py b/src/sage/combinat/tutorial.py
index 878712a7b45..95f07084e97 100644
--- a/src/sage/combinat/tutorial.py
+++ b/src/sage/combinat/tutorial.py
@@ -509,7 +509,7 @@
     sage: bool(equadiff.substitute_function(Cf, s0))
     True
 
-.. On veut non seulement remplacer les occurences de C(z), mais
+.. On veut non seulement remplacer les occurrences de C(z), mais
 .. aussi de C tout court (par exemple dans D[0](C)). Y-a-t'il mieux
 .. pour retrouver C à partir de C(z)?
 .. Cf. also:
diff --git a/src/sage/combinat/words/alphabet.py b/src/sage/combinat/words/alphabet.py
index a7fe2bc414d..bf7a06223eb 100644
--- a/src/sage/combinat/words/alphabet.py
+++ b/src/sage/combinat/words/alphabet.py
@@ -215,7 +215,7 @@ def build_alphabet(data=None, names=None, name=None):
     if name is not None and (data is not None or names is not None):
         raise ValueError("name cannot be specified with any other argument")
 
-    # Swap arguments if we need to to try and make sure we have "good" user input
+    # Swap arguments if we need to try and make sure we have "good" user input
     if isinstance(names, integer_types + (Integer,)) or names == Infinity \
             or (data is None and names is not None):
         data,names = names,data
diff --git a/src/sage/combinat/words/word.py b/src/sage/combinat/words/word.py
index c93e725d27b..d2451061b26 100644
--- a/src/sage/combinat/words/word.py
+++ b/src/sage/combinat/words/word.py
@@ -60,7 +60,7 @@ def Word(data=None, alphabet=None, length=None, datatype=None, caching=True, RSK
        For iterators: Infinity if you know the iterator will not
        terminate (default); ``"unknown"`` if you do not know whether the
        iterator terminates; ``"finite"`` if you know that the iterator
-       terminates, but do know know the length.
+       terminates, but do not know the length.
 
     -  ``datatype`` -- (default: ``None``) ``None``, ``"list"``, ``"str"``,
        ``"tuple"``, ``"iter"``, ``"callable"``. If ``None``, then the function
diff --git a/src/sage/combinat/words/word_char.pyx b/src/sage/combinat/words/word_char.pyx
index 98eb19ade17..beb89e1a5a9 100644
--- a/src/sage/combinat/words/word_char.pyx
+++ b/src/sage/combinat/words/word_char.pyx
@@ -10,7 +10,7 @@ Fast word datatype using an array of unsigned char
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from cysignals.memory cimport check_allocarray, sig_free
 from cysignals.signals cimport sig_on, sig_off
diff --git a/src/sage/combinat/words/words.py b/src/sage/combinat/words/words.py
index 445fa02bfd1..4c82f9b0445 100644
--- a/src/sage/combinat/words/words.py
+++ b/src/sage/combinat/words/words.py
@@ -1879,7 +1879,7 @@ def __call__(self, data=None, length=None, datatype=None, caching=True, check=Tr
            For iterators: Infinity if you know the iterator will not
            terminate (default); "unknown" if you do not know whether the
            iterator terminates; "finite" if you know that the iterator
-           terminates, but do know know the length.
+           terminates, but do not know the length.
 
         -  ``datatype`` - (default: None) None, "char", "list", "str",
            "tuple", "iter", "callable" or "pickled_function". If None, then
diff --git a/src/sage/cpython/dict_del_by_value.pyx b/src/sage/cpython/dict_del_by_value.pyx
index cc8ad667cbd..07eb93860aa 100644
--- a/src/sage/cpython/dict_del_by_value.pyx
+++ b/src/sage/cpython/dict_del_by_value.pyx
@@ -22,7 +22,7 @@ AUTHORS:
 #
 #                  http://www.gnu.org/licenses/
 ########################################################################
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 import weakref
 from weakref import KeyedRef
diff --git a/src/sage/crypto/mq/rijndael_gf.py b/src/sage/crypto/mq/rijndael_gf.py
index 177eace0ea5..36135b76972 100644
--- a/src/sage/crypto/mq/rijndael_gf.py
+++ b/src/sage/crypto/mq/rijndael_gf.py
@@ -444,7 +444,7 @@ def __init__(self, Nb, Nk, state_chr='a', key_chr='k'):
         - ``Nb`` -- The block length of this instantiation. Must be between 4
           and 8.
 
-        - ``Nk`` -- The key length of this instantion. Must be between 4 and 8.
+        - ``Nk`` -- The key length of this instantiation. Must be between 4 and 8.
 
         - ``state_chr`` -- The variable name for polynomials representing
           elements from state matrices.
diff --git a/src/sage/data_structures/bitset.pxi b/src/sage/data_structures/bitset.pxi
index 24b0ef9d258..090b307082b 100644
--- a/src/sage/data_structures/bitset.pxi
+++ b/src/sage/data_structures/bitset.pxi
@@ -85,7 +85,7 @@ cdef inline bint bitset_init(bitset_t bits, mp_bitcnt_t size) except -1:
 
 cdef inline int bitset_realloc(bitset_t bits, mp_bitcnt_t size) except -1:
     """
-    Reallocate a bitset to size size. If reallocation is larger, new bitset
+    Reallocate a bitset to size ``size``. If reallocation is larger, new bitset
     does not contain any of the extra bits.
     """
     cdef mp_size_t limbs_old = bits.limbs
diff --git a/src/sage/databases/jones.py b/src/sage/databases/jones.py
index 1d8f6ad6dd4..ebe4abf06bc 100644
--- a/src/sage/databases/jones.py
+++ b/src/sage/databases/jones.py
@@ -74,6 +74,7 @@
 
 from sage.structure.sage_object import load, save
 
+from sage.misc.package import PackageNotFoundError
 
 JONESDATA = os.path.join(SAGE_SHARE, 'jones')
 
@@ -228,7 +229,7 @@ def get(self, S, var='a'):
             if os.path.exists(JONESDATA + "/jones.sobj"):
                 self.root = load(JONESDATA + "/jones.sobj")
             else:
-                raise RuntimeError("You must install the Jones database optional package.")
+                raise PackageNotFoundError("database_jones_numfield")
         try:
             S = list(S)
         except TypeError:
diff --git a/src/sage/doctest/forker.py b/src/sage/doctest/forker.py
index 6553d96b967..bd02004a481 100644
--- a/src/sage/doctest/forker.py
+++ b/src/sage/doctest/forker.py
@@ -35,6 +35,7 @@
 
 from __future__ import print_function
 from __future__ import absolute_import
+import __future__
 
 import hashlib, multiprocessing, os, sys, time, warnings, signal, linecache
 import doctest, traceback
@@ -45,7 +46,11 @@
 from .parsing import OriginalSource, reduce_hex
 from sage.structure.sage_object import SageObject
 from .parsing import SageOutputChecker, pre_hash, get_source
-from sage.repl.user_globals import set_globals
+from sage.repl.user_globals import set_globals, get_globals
+
+
+# All doctests run as if the following future imports are present
+MANDATORY_COMPILE_FLAGS = __future__.print_function.compiler_flag
 
 
 def init_sage():
@@ -130,6 +135,7 @@ def init_sage():
     stringPict.terminal_width = lambda self:0
 
 
+
 def showwarning_with_traceback(message, category, filename, lineno, file=sys.stdout, line=None):
     r"""
     Displays a warning message with a traceback.
@@ -514,7 +520,13 @@ def _run(self, test, compileflags, out):
             finally:
                 if self.debugger is not None:
                     self.debugger.set_continue() # ==== Example Finished ====
-            got = self._fakeout.getvalue()  # the actual output
+            got = self._fakeout.getvalue()
+            try:
+                got = got.decode('utf-8')
+            except UnicodeDecodeError:
+                got = got.decode('latin1')
+            # the actual output
+
             outcome = FAILURE   # guilty until proved innocent or insane
 
             # If the example executed without raising any exceptions,
@@ -624,6 +636,7 @@ def run(self, test, compileflags=None, out=None, clear_globs=True):
         self.test = test
         if compileflags is None:
             compileflags = doctest._extract_future_flags(test.globs)
+        compileflags |= MANDATORY_COMPILE_FLAGS
         # We use this slightly modified version of Pdb because it
         # interacts better with the doctesting framework (like allowing
         # doctests for sys.settrace()). Since we already have output
@@ -768,7 +781,7 @@ def update_digests(self, example):
             sage: DTR.running_global_digest.hexdigest()
             '3cb44104292c3a3ab4da3112ce5dc35c'
         """
-        s = pre_hash(get_source(example))
+        s = pre_hash(get_source(example)).encode('utf-8')
         self.running_global_digest.update(s)
         self.running_doctest_digest.update(s)
         if example.predecessors is not None:
diff --git a/src/sage/doctest/parsing.py b/src/sage/doctest/parsing.py
index 77a1c8f8488..01746f7f558 100644
--- a/src/sage/doctest/parsing.py
+++ b/src/sage/doctest/parsing.py
@@ -22,8 +22,9 @@
 #  the License, or (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function, absolute_import
+from __future__ import print_function, absolute_import, unicode_literals
 from sage.misc.six import u
+from six import text_type
 
 import re
 import sys
@@ -291,7 +292,7 @@ def reduce_hex(fingerprints):
     return "%032x" % res
 
 
-class MarkedOutput(str):
+class MarkedOutput(text_type):
     """
     A subclass of string with context for whether another string
     matches it.
@@ -306,6 +307,9 @@ class MarkedOutput(str):
         'abc'
         sage: s.rel_tol
         0.0500000000000000
+
+        sage: MarkedOutput(u"56 µs")
+        u'56 \xb5s'
     """
     random = False
     rel_tol = 0
diff --git a/src/sage/doctest/sources.py b/src/sage/doctest/sources.py
index c8bd433e4c7..f3e05a6654e 100644
--- a/src/sage/doctest/sources.py
+++ b/src/sage/doctest/sources.py
@@ -19,7 +19,7 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 from __future__ import print_function
-from __future__ import absolute_import
+from __future__ import absolute_import, unicode_literals
 
 import os
 import sys
@@ -61,6 +61,9 @@
 sagestart = re.compile(r"^\s*(>>> |sage: )\s*[^#\s]")
 untested = re.compile("(not implemented|not tested)")
 
+# For parsing a PEP 0263 encoding declaration
+pep_0263 = re.compile(r'coding[:=]\s*([-\w.]+)')
+
 # Source line number in warning output
 doctest_line_number = re.compile(r"^\s*doctest:[0-9]")
 
@@ -226,7 +229,7 @@ def _process_doc(self, doctests, doc, namespace, start):
 
     def _create_doctests(self, namespace, tab_okay=None):
         """
-        Creates a list doctests defined in this source.
+        Creates a list of doctests defined in this source.
 
         This function collects functionality common to file and string
         sources, and is called by
@@ -511,10 +514,13 @@ def __init__(self, path, options):
         base, ext = os.path.splitext(path)
         if ext in ('.py', '.pyx', '.pxd', '.pxi', '.sage', '.spyx'):
             self.__class__ = dynamic_class('PythonFileSource',(FileDocTestSource,PythonSource))
+            self.encoding = "utf-8"
         elif ext == '.tex':
             self.__class__ = dynamic_class('TexFileSource',(FileDocTestSource,TexSource))
+            self.encoding = "utf-8"
         elif ext == '.rst':
             self.__class__ = dynamic_class('RestFileSource',(FileDocTestSource,RestSource))
+            self.encoding = "utf-8"
         else:
             raise ValueError("unknown file extension %r"%ext)
 
@@ -536,10 +542,38 @@ def __iter__(self):
             1     sage: 2 + 2
             2     4
             3 '''
+
+        The encoding is "utf-8" by default::
+
+            sage: FDS.encoding
+            'utf-8'
+
+        We create a file with a Latin-1 encoding without declaring it::
+
+            sage: s = "'''\nRegardons le polyn\xF4me...\n'''\n"
+            sage: open(filename, 'w').write(s)
+            sage: FDS = FileDocTestSource(filename, DocTestDefaults())
+            sage: L = list(FDS)
+            Traceback (most recent call last):
+            ...
+            UnicodeDecodeError: 'utf8' codec can't decode byte 0xf4 in position 18: invalid continuation byte
+
+        This works if we add a PEP 0263 encoding declaration::
+
+            sage: s = "#!/usr/bin/env python\n# -*- coding: latin-1 -*-\n" + s
+            sage: open(filename, 'w').write(s)
+            sage: FDS = FileDocTestSource(filename, DocTestDefaults())
+            sage: L = list(FDS)
+            sage: FDS.encoding
+            'latin-1'
         """
         with open(self.path) as source:
             for lineno, line in enumerate(source):
-                yield lineno, line
+                if lineno < 2:
+                    match = pep_0263.search(line)
+                    if match:
+                        self.encoding = match.group(1)
+                yield lineno, unicode(line, self.encoding)
 
     @lazy_attribute
     def printpath(self):
diff --git a/src/sage/dynamics/complex_dynamics/all.py b/src/sage/dynamics/complex_dynamics/all.py
index 8106e7b4253..02b2d44fc78 100644
--- a/src/sage/dynamics/complex_dynamics/all.py
+++ b/src/sage/dynamics/complex_dynamics/all.py
@@ -1,4 +1,4 @@
 from __future__ import absolute_import
 from sage.misc.lazy_import import lazy_import
-lazy_import("sage.dynamics.complex_dynamics.mandel_julia", "mandelbrot_plot")
-lazy_import("sage.dynamics.complex_dynamics.mandel_julia", "external_ray")
+lazy_import("sage.dynamics.complex_dynamics.mandel_julia",
+            ["mandelbrot_plot", "external_ray", "julia_plot"])
diff --git a/src/sage/dynamics/complex_dynamics/mandel_julia.py b/src/sage/dynamics/complex_dynamics/mandel_julia.py
index 3157fc3dad5..15249e4a754 100644
--- a/src/sage/dynamics/complex_dynamics/mandel_julia.py
+++ b/src/sage/dynamics/complex_dynamics/mandel_julia.py
@@ -1,13 +1,17 @@
 r"""
 Mandelbrot and Julia sets
 
-Plots the Mandelbrot set for the map `Q_c(z)=z^2+c` in the complex plane.
+Plots the Mandelbrot and Julia sets for the map `Q_c(z)=z^2+c` in the complex
+plane.
 
 The Mandelbrot set is the set of complex numbers `c` for which the function
 `Q_c(z)=z^2+c` does not diverge when iterated from `z = 0`. This set of complex
 numbers can be visualized by plotting each value for `c` in the complex plane.
 The Mandelbrot set is an example of a fractal when plotted in the complex plane.
 
+The Julia set for a given `c` is the set of complex numbers for which the
+function `Q_c(z)=z^2+c` is bounded under iteration.
+
 AUTHORS:
 
 - Ben Barros
@@ -25,13 +29,26 @@
 #*****************************************************************************
 
 from __future__ import absolute_import, division
-from sage.dynamics.complex_dynamics.mandel_julia_helper import \
- fast_mandelbrot_plot, fast_external_ray, convert_to_pixels, get_line
-from sagenb.notebook.interact import interact, slider, input_box, color_selector
+from sage.dynamics.complex_dynamics.mandel_julia_helper import (fast_mandelbrot_plot,
+                                                                fast_external_ray,
+                                                                convert_to_pixels,
+                                                                get_line,
+                                                                fast_julia_plot,
+                                                                julia_helper)
+from sagenb.notebook.interact import (interact,
+                                      slider,
+                                      input_box,
+                                      color_selector,
+                                      checkbox)
 from sage.plot.colors import Color
 from sage.repl.image import Image
 from sage.functions.log import function_log as log
-
+from sage.rings.rational_field import QQ
+from sage.rings.all import CC
+from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+from sage.schemes.projective.projective_space import ProjectiveSpace
+from sage.categories.homset import End
+from sage.misc.prandom import randint
 
 def mandelbrot_plot(**kwds):
     r"""
@@ -127,8 +144,8 @@ def _(real_center=input_box(x_center, 'Real'),
             color_num=input_box(number_of_colors, 'Number of Colors'),
             image_color=color_selector(default=Color([j/255 for j in base_color]),
              label="Image Color", hide_box=True)):
-            return fast_mandelbrot_plot(real_center, im_center, width, iterations,
-             pixel_count, level_sep, color_num, image_color).show()
+            return fast_mandelbrot_plot(real_center, im_center, width,
+             iterations, pixel_count, level_sep, color_num, image_color).show()
 
     else:
         return fast_mandelbrot_plot(x_center, y_center, image_width, max_iteration,
@@ -268,3 +285,174 @@ def external_ray(theta, **kwds):
             if max(k) < pixel_width and min(k) >= 0:
                 pixel[int(k[0]), int(k[1])] = tuple(ray_color)
     return M
+
+def julia_plot(c=-1, **kwds):
+    r"""
+    Plots the Julia set of a given complex `c` value. Users can specify whether
+    they would like to display the Mandelbrot side by side with the Julia set.
+
+    The Julia set of a given `c` value is the set of complex numbers for which
+    the function `Q_c(z)=z^2+c` is bounded under iteration. The Julia set can
+    be visualized by plotting each point in the set in the complex plane.
+    Julia sets are examples of fractals when plotted in the complex plane.
+
+    ALGORITHM:
+
+    Define the map `Q_c(z) = z^2 + c` for some `c \in \mathbb{C}`. For every
+    `p \in \mathbb{C}`, if `|Q_{c}^{k}(p)| > 2` for some `k \geq 0`,
+    then `Q_{c}^{n}(p) \to \infty`. Let `N` be the maximum number of iterations.
+    Compute the first `N` points on the orbit of `p` under `Q_c`. If for
+    any `k < N`, `|Q_{c}^{k}(p)| > 2`, we stop the iteration and assign a color
+    to the point `p` based on how quickly `p` escaped to infinity under
+    iteration of `Q_c`. If `|Q_{c}^{i}(p)| \leq 2` for all `i \leq N`, we assume
+    `p` is in the Julia set and assign the point `p` the color black.
+
+    INPUT:
+
+    - ``c`` -- complex (optional - default: ``-1``), complex point `c` that
+      determines the Julia set.
+
+    kwds:
+
+    - ``period`` -- list (optional - default: ``None``), returns the Julia set
+      for a random `c` value with the given (formal) cycle structure.
+
+    - ``mandelbrot`` -- boolean (optional - default: ``True``), when set to
+      ``True``, an image of the Mandelbrot set is appended to the right of the
+      Julia set.
+
+    - ``point_color`` -- RGB color (optional - default: ``[255, 0, 0]``),
+      color of the point `c` in the Mandelbrot set.
+
+    - ``x_center`` -- double (optional - default: ``-1.0``), Real part
+      of center point.
+
+    - ``y_center`` -- double (optional - default: ``0.0``), Imaginary part
+      of center point.
+
+    - ``image_width`` -- double (optional - default: ``4.0``), width of image
+      in the complex plane.
+
+    - ``max_iteration`` -- long (optional - default: ``500``), maximum number
+      of iterations the map `Q_c(z)`.
+
+    - ``pixel_count`` -- long (optional - default: ``500``), side length of
+      image in number of pixels.
+
+    - ``base_color`` -- RGB color (optional - default: ``[40, 40, 40]``), color
+      used to determine the coloring of set.
+
+    - ``iteration_level`` -- long (optional - default: 1), number of iterations
+      between each color level.
+
+    - ``number_of_colors`` -- long (optional - default: 30), number of colors
+      used to plot image.
+
+    - ``interact`` -- boolean (optional - default: ``False``), controls whether
+      plot will have interactive functionality.
+
+    OUTPUT:
+
+    24-bit RGB image of the Julia set in the complex plane.
+
+    EXAMPLES::
+
+        sage: julia_plot()
+        1001x500px 24-bit RGB image
+
+    To display only the Julia set, set ``mandelbrot`` to ``False``::
+
+        sage: julia_plot(mandelbrot=False)
+        500x500px 24-bit RGB image
+
+    To display an interactive plot of the Julia set in the Notebook,
+    set ``interact`` to ``True``::
+
+        sage: julia_plot(interact=True)
+        <html>...</html>
+
+    To return the Julia set of a random `c` value with (formal) cycle structure
+    `(2,3)`, set ``period = [2,3]``::
+
+        sage: julia_plot(period=[2,3])
+        1001x500px 24-bit RGB image
+
+    To return all of the Julia sets of `c` values with (formal) cycle structure
+    `(2,3)`::
+
+        sage: period = [2,3] # not tested
+        ....: R.<c> = QQ[]
+        ....: P.<x,y> = ProjectiveSpace(R,1)
+        ....: R = P.coordinate_ring()
+        ....: H = End(P)
+        ....: f = H([x^2+c*y^2,y^2])
+        ....: L = f.dynatomic_polynomial(period).subs({x:0,y:1}).roots(ring=CC)
+        ....: c_values = [k[0] for k in L]
+        ....: for c in c_values:
+        ....:     julia_plot(c)
+    """
+
+    x_center = kwds.pop("x_center", 0.0)
+    y_center = kwds.pop("y_center", 0.0)
+    image_width = kwds.pop("image_width", 4.0)
+    max_iteration = kwds.pop("max_iteration", 500)
+    pixel_count = kwds.pop("pixel_count", 500)
+    base_color = kwds.pop("base_color", [50, 50, 50])
+    iteration_level = kwds.pop("iteration_level", 1)
+    number_of_colors = kwds.pop("number_of_colors", 50)
+    point_color = kwds.pop("point_color", [255, 0, 0])
+    interacts = kwds.pop("interact", False)
+    mandelbrot = kwds.pop("mandelbrot", True)
+    period = kwds.pop("period", None)
+
+    if not period is None:
+        R = PolynomialRing(QQ, 'c')
+        c = R.gen()
+        P = ProjectiveSpace(R, 1, 'x,y')
+        x,y = P.gens()
+        H = End(P)
+        f = H([x**2+c*y**2, y**2])
+        L = f.dynatomic_polynomial(period).subs({x:0,y:1}).roots(ring=CC)
+        c = L[randint(0,len(L)-1)][0]
+
+    c_real = CC(c).real()
+    c_imag = CC(c).imag()
+
+    if interacts:
+        @interact(layout={'bottom':[['real_center'], ['im_center'], ['width']],
+         'top':[['iterations'], ['level_sep'], ['color_num'], ['mandel'],
+         ['cx'], ['cy']], 'right':[['image_color'], ['pt_color']]})
+        def _(cx = input_box(c_real, '$Re(c)$'),
+            cy = input_box(c_imag, '$Im(c)$'),
+            real_center=input_box(x_center, 'Real Center'),
+            im_center=input_box(y_center, 'Imaginary Center'),
+            width=input_box(image_width, 'Width of Image'),
+            iterations=input_box(max_iteration, 'Max Number of Iterations'),
+            level_sep=input_box(iteration_level, 'Iterations between Colors'),
+            color_num=input_box(number_of_colors, 'Number of Colors'),
+            image_color=color_selector(default=Color([j/255 for j in base_color]),
+             label="Image Color", hide_box=True),
+            pt_color=color_selector(default=Color([j/255 for j in point_color]),
+             label="Point Color", hide_box=True),
+            mandel=checkbox(mandelbrot, label='Mandelbrot set')):
+
+            if mandel:
+                return julia_helper(cx, cy, real_center, im_center,
+                 width, iterations, pixel_count, level_sep, color_num,
+                 image_color, pt_color).show()
+
+            else:
+                return fast_julia_plot(cx, cy, real_center, im_center,
+                 width, iterations, pixel_count, level_sep, color_num,
+                 image_color).show()
+
+    else:
+        if mandelbrot:
+            return julia_helper(c_real, c_imag, x_center, y_center,
+             image_width, max_iteration, pixel_count, iteration_level,
+             number_of_colors, base_color, point_color)
+
+        else:
+            return fast_julia_plot(c_real, c_imag, x_center, y_center,
+             image_width, max_iteration, pixel_count, iteration_level,
+             number_of_colors, base_color)
diff --git a/src/sage/dynamics/complex_dynamics/mandel_julia_helper.pyx b/src/sage/dynamics/complex_dynamics/mandel_julia_helper.pyx
index fc954a46c33..85f1f3e4ef8 100644
--- a/src/sage/dynamics/complex_dynamics/mandel_julia_helper.pyx
+++ b/src/sage/dynamics/complex_dynamics/mandel_julia_helper.pyx
@@ -256,7 +256,7 @@ cpdef convert_to_pixels(point_list, double x_0, double y_0, double width,
 
     OUTPUT:
 
-    List of tuples of integers representing the pixels of the external ray.
+    List of tuples of integers representing pixels.
 
     EXAMPLES::
 
@@ -359,3 +359,210 @@ cpdef get_line(start, end):
     if swapped:
         points.reverse()
     return points
+
+cpdef fast_julia_plot(double c_real, double c_imag,
+  double x_center=0, double y_center=0, double image_width=4,
+  long max_iteration=500, long pixel_count=500, long level_sep=2,
+  long color_num=40, base_color=[50, 50, 50]):
+    r"""
+    Plots the Julia set for a given `c` value in the complex plane for the map `Q_c(z) = z^2 + c`.
+
+    INPUT:
+
+    - ``c_real`` -- double, Real part of `c` value that determines Julia set.
+
+    - ``c_imag`` -- double, Imaginary part of `c` value that determines Julia set.
+
+    - ``x_center`` -- double (optional - default: ``0.0``), Real part of center point.
+
+    - ``y_center`` -- double (optional - default: ``0.0``), Imaginary part of center point.
+
+    - ``image_width`` -- double (optional - default: ``4.0``), width of image in the complex plane.
+
+    - ``max_iteration`` -- long (optional - default: ``500``), maximum number of iterations the map ``Q_c(z)``.
+
+    - ``pixel_count`` -- long (optional - default: ``500``), side length of image in number of pixels.
+
+    - ``level_sep`` -- long (optional - default: ``2``), number of iterations between each color level.
+
+    - ``color_num`` -- long (optional - default: ``40``), number of colors used to plot image.
+
+    - ``base_color`` -- RGB color (optional - default: ``[50, 50, 50]``), color used to determine the coloring of set.
+
+    OUTPUT:
+
+    24-bit RGB image of the Julia set in the complex plane.
+
+    EXAMPLES:
+
+    Plot the Julia set for `c=-1+0i`::
+
+        sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_julia_plot
+        sage: fast_julia_plot(-1, 0)
+        500x500px 24-bit RGB image
+
+    We can focus on smaller parts of the set by adjusting ``image_width``::
+
+        sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_julia_plot
+        sage: fast_julia_plot(-0.7, 0.3, x_center=.75, image_width=0.01, color_num=100)
+        500x500px 24-bit RGB image
+    """
+
+    cdef long i, j, col, row, level, color_value, iteration
+    cdef double k, x_corner, y_corner, step_size, x_coor, y_coor, new_x, new_y
+    cdef M, pixel, color_list
+
+    # Make sure image_width is positive
+    image_width = abs(image_width)
+
+    # Initialize an image to the color black and access the pixels
+    J = Image("RGB", (pixel_count,pixel_count), 'black')
+    Jp = J.pixels()
+
+    # Take the given base color and create a list of evenly spaced
+    # colors between the given base color and white. The number of
+    # colors in the list depends on the variable color_num.
+    if type(base_color) == Color:
+        # Convert Color to RGB list
+        base_color = [int(k*255) for k in base_color]
+    color_list = []
+    for i in range(color_num):
+        sig_check()
+        color_list.append(copy(base_color))
+        for j in range(3):
+            color_list[i][j] += i * (255 - color_list[i][j]) // color_num
+        color_list[i] = tuple(color_list[i])
+
+    # First, we determine the complex coordinates of the point in the top left
+    # corner of the image. Then, we loop through each pixel in the image and
+    # assign it complex coordinates relative to the image's top left corner.
+    x_corner = x_center - image_width/2
+    y_corner = y_center + image_width/2
+    step_size = image_width / pixel_count
+    for col in range(pixel_count):
+        x_coor = x_corner + col*step_size
+        for row in range(pixel_count):
+            sig_check()
+            y_coor = y_corner - row*step_size
+
+            # We compute the orbit of each pixel under the map Q(z) = z^2 + c
+            # until we either reach the maximum number of iterations
+            # or find a point in the orbit with modulus greater than 2
+            new_x, new_y = x_coor, y_coor
+            iteration = 0
+            while (new_x**2 + new_y**2 <= 4.0 and iteration < max_iteration):
+                sig_check()
+                new_x, new_y = new_x**2 - new_y**2 + c_real, \
+                 2*new_x*new_y + c_imag
+                iteration += 1
+
+            # If the point escapes to infinity, assign the point a color
+            # based on how fast it escapes. The more iterations it takes for
+            # a point to escape to infinity, the lighter its color will be.
+            # Otherwise, assume the point is in the Julia set and leave
+            # it black.
+            if iteration != max_iteration:
+                # Assign each point a level based on its number of iterations.
+                level = iteration // level_sep
+                # Assign the pixel a color based on it's level. If we run out
+                # of colors, assign it the last color in the list.
+                if level < color_num:
+                    Jp[col,row] = color_list[level]
+                else:
+                    Jp[col,row] = color_list[-1]
+
+    return J
+
+cpdef julia_helper(double c_real, double c_imag, double x_center=0,
+ double y_center=0, double image_width=4, long max_iteration=500,
+ long pixel_count=500, long level_sep=2, long color_num=40,
+ base_color=[50, 50, 50], point_color=[255, 0, 0]):
+
+    r"""
+    Helper function that returns the image of a Julia set for a given
+    `c` value side by side with the Mandelbrot set with a point denoting
+    the `c` value.
+
+    INPUT:
+
+    - ``c_real`` -- double, Real part of `c` value that determines Julia set.
+
+    - ``c_imag`` -- double, Imaginary part of `c` value that determines Julia set.
+
+    - ``x_center`` -- double (optional - default: ``0.0``), Real part of center point.
+
+    - ``y_center`` -- double (optional - default: ``0.0``), Imaginary part of center point.
+
+    - ``image_width`` -- double (optional - default: ``4.0``), width of image in the complex plane.
+
+    - ``max_iteration`` -- long (optional - default: ``500``), maximum number of iterations the map ``Q_c(z)``.
+
+    - ``pixel_count`` -- long (optional - default: ``500``), side length of image in number of pixels.
+
+    - ``level_sep`` -- long (optional - default: ``2``), number of iterations between each color level.
+
+    - ``color_num`` -- long (optional - default: ``40``), number of colors used to plot image.
+
+    - ``base_color`` -- RGB color (optional - default: ``[50, 50, 50]``), color used to determine the coloring of set.
+
+    - ``point_color`` -- RGB color (optional - default: ``[255, 0, 0]``), color of the point `c` in the Mandelbrot set.
+
+    OUTPUT:
+
+    24-bit RGB image of the Julia and Mandelbrot sets in the complex plane.
+
+    EXAMPLES:
+
+    Plot the Julia set for `c=-1+0i`::
+
+        sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import julia_helper
+        sage: julia_helper(-1,0)
+        1001x500px 24-bit RGB image
+
+    We can focus on smaller parts of the set by adjusting ``image_width``::
+
+        sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import julia_helper
+        sage: julia_helper(-.5, .6, y_center=0.178, image_width=0.01) # long time
+        1001x500px 24-bit RGB image
+    """
+
+    cdef:
+        int i, j
+        M, Mp, G, Gp, J, Jp, CP
+
+    # Initialize the Julia set
+    J = fast_julia_plot(c_real, c_imag, x_center, y_center, image_width,
+     max_iteration, pixel_count, level_sep, color_num, base_color)
+    Jp = J.pixels()
+
+    # Initialize the image with Julia set on left side
+    # Add white border between images
+    G = Image("RGB", (2*pixel_count+1,pixel_count), 'white')
+    Gp = G.pixels()
+    for i in range(pixel_count):
+        for j in range(pixel_count):
+            Gp[i,j] = Jp[i,j]
+
+    # Plot the Mandelbrot set on the right side
+    M = fast_mandelbrot_plot(-1, 0, 4, 500, pixel_count, 1, 30, base_color)
+    Mp = M.pixels()
+    for i in range(pixel_count+1,2*pixel_count):
+        for j in range(pixel_count):
+            Gp[i,j] = Mp[int(i-pixel_count),j]
+
+    # Convert Color to RGB list if necessary
+    if type(point_color) == Color:
+        point_color = [int(k*255) for k in point_color]
+
+    # Add a cross representing c-value to the Mandelbrot set.
+    CP = convert_to_pixels([(c_real, c_imag)], -1, 0, 4, pixel_count)
+    for i in range(-3,4):
+        # Loop through x and y coordinates and check if they are in image
+        if min(CP[0][0]+i, CP[0][1]) >= 0 and \
+         max(CP[0][0]+i, CP[0][1]) < pixel_count:
+            Gp[CP[0][0]+i+pixel_count+1, CP[0][1]] = tuple(point_color)
+        if min(CP[0][0], CP[0][1]+i) >= 0 and \
+         max(CP[0][0], CP[0][1]+i) < pixel_count:
+            Gp[CP[0][0]+pixel_count+1, CP[0][1]+i] = tuple(point_color)
+
+    return G
diff --git a/src/sage/dynamics/flat_surfaces/strata.py b/src/sage/dynamics/flat_surfaces/strata.py
index d7c8ed2737e..4bd8c726cd7 100644
--- a/src/sage/dynamics/flat_surfaces/strata.py
+++ b/src/sage/dynamics/flat_surfaces/strata.py
@@ -664,7 +664,7 @@ class AbelianStratum(SageObject):
     :meth:`sage.dynamics.interval_exchanges.template.Permutation.symmetric`
     operation of Boissy-Lanneau).
 
-    When you want to specify a marked separatrix, the degree on which it is is
+    When you want to specify a marked separatrix, the degree on which it is
     the first term of your degrees list.
 
     INPUT:
diff --git a/src/sage/dynamics/interval_exchanges/iet.py b/src/sage/dynamics/interval_exchanges/iet.py
index 7ab82674607..a6e7f780be4 100644
--- a/src/sage/dynamics/interval_exchanges/iet.py
+++ b/src/sage/dynamics/interval_exchanges/iet.py
@@ -526,7 +526,7 @@ def in_which_interval(self, x, interval=0):
 
         label -- a label corresponding to an interval
 
-        TEST:
+        TESTS:
 
         ::
 
diff --git a/src/sage/ext/fast_callable.pyx b/src/sage/ext/fast_callable.pyx
index 2e4ea16967a..7ce49e76a21 100644
--- a/src/sage/ext/fast_callable.pyx
+++ b/src/sage/ext/fast_callable.pyx
@@ -299,7 +299,7 @@ AUTHOR:
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-
+from __future__ import absolute_import
 
 
 import operator
diff --git a/src/sage/game_theory/catalog_normal_form_games.py b/src/sage/game_theory/catalog_normal_form_games.py
index c0a8aa88587..e60fe48a4db 100644
--- a/src/sage/game_theory/catalog_normal_form_games.py
+++ b/src/sage/game_theory/catalog_normal_form_games.py
@@ -331,7 +331,7 @@ def AntiCoordinationGame(A=3, a=3, B=5, b=1, C=1, c=5, D=0, d=0):
     Return a 2 by 2 AntiCoordination Game.
 
     An anti coordination game is a particular type of game where the pure Nash
-    equilibria is for the players to pick different strategies strategies.
+    equilibria is for the players to pick different strategies.
 
     In general these are represented as a normal form game using the
     following two matrices:
diff --git a/src/sage/game_theory/cooperative_game.py b/src/sage/game_theory/cooperative_game.py
index 6e292d6b4d6..2ec0d8ae5a7 100644
--- a/src/sage/game_theory/cooperative_game.py
+++ b/src/sage/game_theory/cooperative_game.py
@@ -61,7 +61,7 @@ class CooperativeGame(SageObject):
         42 &\text{if } c = \{1,2,3\}. \\
         \end{cases}
 
-    The function `v` can be thought of as as a record of contribution of
+    The function `v` can be thought of as a record of contribution of
     individuals and coalitions of individuals. Of interest, becomes how to
     fairly share the value of the grand coalition (`\Omega`)? This class
     allows for such an answer to be formulated by calculating the Shapley
diff --git a/src/sage/game_theory/normal_form_game.py b/src/sage/game_theory/normal_form_game.py
index eb3f7cc55e3..9d6d0be17cb 100644
--- a/src/sage/game_theory/normal_form_game.py
+++ b/src/sage/game_theory/normal_form_game.py
@@ -603,7 +603,7 @@
 from sage.structure.sage_object import SageObject
 from sage.matrix.constructor import matrix
 from sage.matrix.constructor import vector
-from sage.misc.package import is_package_installed
+from sage.misc.package import is_package_installed, PackageNotFoundError
 from sage.misc.temporary_file import tmp_filename
 
 try:
@@ -1313,13 +1313,13 @@ def obtain_nash(self, algorithm=False, maximization=True):
 
         if algorithm == "lrs":
             if not is_package_installed('lrslib'):
-                raise NotImplementedError("lrslib is not installed")
+                raise PackageNotFoundError("lrslib")
 
             return self._solve_lrs(maximization)
 
         if algorithm == "LCP":
             if Game is None:
-                raise NotImplementedError("gambit is not installed")
+                raise PackageNotFoundError("gambit")
             for strategy_profile in self.utilities:
                 payoffs = self.utilities[strategy_profile]
                 if payoffs != [int(payoffs[0]), int(payoffs[1])]:
diff --git a/src/sage/games/sudoku.py b/src/sage/games/sudoku.py
index dd1cbc8fb23..d1b37c663d6 100644
--- a/src/sage/games/sudoku.py
+++ b/src/sage/games/sudoku.py
@@ -366,7 +366,7 @@ def to_list(self):
             sage: s.to_list()
             [1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 9, 0, 4, 0, 0, 0, 5, 0, 0, 0, 6, 0, 0, 0, 7, 0, 0, 0, 5, 0, 9, 0, 3, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 8, 5, 0, 0, 4, 0, 7, 0, 0, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0, 9, 0, 8, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1]
 
-        TEST:
+        TESTS:
 
         This tests the input and output of Sudoku puzzles as lists. ::
 
@@ -389,7 +389,7 @@ def to_matrix(self):
             [0 0 1 4]
             [0 0 3 0]
 
-        TEST:
+        TESTS:
 
         This tests the input and output of Sudoku puzzles as matrices over `\ZZ`. ::
 
diff --git a/src/sage/games/sudoku_backtrack.pyx b/src/sage/games/sudoku_backtrack.pyx
index e660ff4430c..ad28e31c150 100644
--- a/src/sage/games/sudoku_backtrack.pyx
+++ b/src/sage/games/sudoku_backtrack.pyx
@@ -20,7 +20,7 @@ def backtrack_all(n, puzzle):
 
         A list of solutions, where each solution is a (1-based) list similar to ``puzzle``.
 
-    TEST:
+    TESTS:
 
     This is just a cursory test here, since eventually this code will move.
     See the `backtrack` method of the `Sudoku` class in the
diff --git a/src/sage/geometry/cone.py b/src/sage/geometry/cone.py
index 68048d6e97f..9ce5337bfdc 100644
--- a/src/sage/geometry/cone.py
+++ b/src/sage/geometry/cone.py
@@ -2315,7 +2315,7 @@ def face_lattice(self):
             if self._ambient is self:
                 # We need to compute face lattice on our own. To accommodate
                 # non-strictly convex cones we split rays (or rather their
-                # indicies) into those in the linear subspace and others, which
+                # indices) into those in the linear subspace and others, which
                 # we refer to as atoms.
                 S = self.linear_subspace()
                 subspace_rays = []
@@ -3720,7 +3720,7 @@ def relative_quotient(self, subcone):
         ``self.sublattice()``. This method returns the quotient
         lattice. The lifts of the quotient generators are
         `\dim(\sigma)-\dim(\rho)` linearly independent primitive
-        lattice lattice points that, together with `N_\rho`, generate
+        lattice points that, together with `N_\rho`, generate
         `N_\sigma`.
 
         OUTPUT:
diff --git a/src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py b/src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py
index b88ae461ad5..1ea867ad453 100644
--- a/src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py
+++ b/src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py
@@ -724,7 +724,7 @@ def complete(self):
             sage: g.is_complete()
             True
 
-        TEST:
+        TESTS:
 
         Check that floating points remain floating points through this method::
 
@@ -975,7 +975,7 @@ def dist(self, other):
             sage: g.dist(p)
             +Infinity
 
-        TEST:
+        TESTS:
 
         Check that floating points remain floating points in :meth:`dist` ::
 
@@ -1646,7 +1646,7 @@ def _to_std_geod(self, p):
             sage: bool(abs(A(e).coordinates()) > 10**9)
             True
 
-        TEST:
+        TESTS:
 
         Check that floating points remain floating points through this method::
 
diff --git a/src/sage/geometry/integral_points.pyx b/src/sage/geometry/integral_points.pyx
index 7a271256c03..111f0af62e3 100644
--- a/src/sage/geometry/integral_points.pyx
+++ b/src/sage/geometry/integral_points.pyx
@@ -12,7 +12,7 @@ Cython helper methods to compute integral points in polyhedra.
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from cysignals.signals cimport sig_check
 import copy
diff --git a/src/sage/geometry/lattice_polytope.py b/src/sage/geometry/lattice_polytope.py
index 54c2bb6445d..57dd21273fc 100644
--- a/src/sage/geometry/lattice_polytope.py
+++ b/src/sage/geometry/lattice_polytope.py
@@ -3437,7 +3437,7 @@ def plot3d(self,
            vertex labels
 
         -  ``vlabels`` - (default:None) if specified, must be a list of labels
-           for each vertex, default labels are vertex indicies
+           for each vertex, default labels are vertex indices
 
         -  ``show_pindices`` - (default:same as show_points)
            whether to show indices of other points
diff --git a/src/sage/geometry/polyhedron/base.py b/src/sage/geometry/polyhedron/base.py
index bb2238e4e71..9e7ff965eec 100644
--- a/src/sage/geometry/polyhedron/base.py
+++ b/src/sage/geometry/polyhedron/base.py
@@ -22,7 +22,7 @@
 from sage.structure.richcmp import rich_to_bool, op_NE
 
 from sage.misc.all import cached_method, prod
-from sage.misc.package import is_package_installed
+from sage.misc.package import is_package_installed, PackageNotFoundError
 
 from sage.rings.all import QQ, ZZ, AA
 from sage.rings.real_double import RDF
@@ -4189,8 +4189,7 @@ def _volume_lrs(self, verbose=False):
              David Avis's lrs program.
         """
         if not is_package_installed('lrslib'):
-            raise NotImplementedError('You must install the optional lrslib package '
-                                       'for this function to work')
+            raise PackageNotFoundError('lrslib')
 
         from sage.misc.temporary_file import tmp_filename
         from subprocess import Popen, PIPE
@@ -4279,7 +4278,7 @@ def _volume_latte(self, verbose=False, algorithm='triangulate', **kwargs):
                 return integrate(self.cdd_Hrepresentation(), algorithm=algorithm, cdd=True, verbose=verbose, **kwargs)
 
         else:
-            raise NotImplementedError('You must install the optional latte_int package for this function to work.')
+            raise PackageNotFoundError('latte_int')
 
     @cached_method
     def volume(self, engine='auto', **kwds):
@@ -4444,7 +4443,7 @@ def integrate(self, polynomial, **kwds):
                 return integrate(self.cdd_Hrepresentation(), polynomial, cdd=True)
 
         else:
-            raise NotImplementedError('You must install the optional latte_int package for this function to work.')
+            raise PackageNotFoundError('latte_int')
 
     def contains(self, point):
         """
diff --git a/src/sage/graphs/asteroidal_triples.pyx b/src/sage/graphs/asteroidal_triples.pyx
index 73d2f9899eb..38e87a2c6fe 100644
--- a/src/sage/graphs/asteroidal_triples.pyx
+++ b/src/sage/graphs/asteroidal_triples.pyx
@@ -127,7 +127,7 @@ def is_asteroidal_triple_free(G, certificate=False):
         sage: is_asteroidal_triple_free(G, certificate=True)
         (False, [0, 2, 6])
 
-    TEST:
+    TESTS:
 
     Giving anything else than a Graph::
 
diff --git a/src/sage/graphs/base/boost_graph.pyx b/src/sage/graphs/base/boost_graph.pyx
index 796c966b10b..3be998c6ce5 100644
--- a/src/sage/graphs/base/boost_graph.pyx
+++ b/src/sage/graphs/base/boost_graph.pyx
@@ -46,6 +46,7 @@ Functions
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
+cimport cython
 from cysignals.signals cimport sig_check, sig_on, sig_off
 
 
@@ -294,6 +295,7 @@ cpdef clustering_coeff(g, vertices=None):
     return (average_clustering, clust_v_sage)
 
 
+@cython.binding(True)
 cpdef dominator_tree(g, root, return_dict=False):
     r"""
     Uses Boost to compute the dominator tree of ``g``, rooted at ``root``.
diff --git a/src/sage/graphs/base/c_graph.pyx b/src/sage/graphs/base/c_graph.pyx
index d81b59debad..13737577bd5 100644
--- a/src/sage/graphs/base/c_graph.pyx
+++ b/src/sage/graphs/base/c_graph.pyx
@@ -2257,7 +2257,7 @@ cdef class CGraphBackend(GenericGraphBackend):
             sage: G.shortest_path_length(1, 3, weight_function=lambda e:e[2]['weight'])
             2
 
-        TEST:
+        TESTS:
 
         Bugfix from :trac:`7673` ::
 
diff --git a/src/sage/graphs/base/static_dense_graph.pyx b/src/sage/graphs/base/static_dense_graph.pyx
index 963d25970f2..a073d6dcfd4 100644
--- a/src/sage/graphs/base/static_dense_graph.pyx
+++ b/src/sage/graphs/base/static_dense_graph.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 r"""
 Static dense graphs
 
diff --git a/src/sage/graphs/base/static_sparse_graph.pyx b/src/sage/graphs/base/static_sparse_graph.pyx
index b3aa2f73dff..8e692cb9968 100644
--- a/src/sage/graphs/base/static_sparse_graph.pyx
+++ b/src/sage/graphs/base/static_sparse_graph.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 r"""
 Static Sparse Graphs
 
diff --git a/src/sage/graphs/chrompoly.pyx b/src/sage/graphs/chrompoly.pyx
index c5027ae830b..f82eb6fa562 100644
--- a/src/sage/graphs/chrompoly.pyx
+++ b/src/sage/graphs/chrompoly.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 """
 Chromatic Polynomial
 
diff --git a/src/sage/graphs/cliquer.pyx b/src/sage/graphs/cliquer.pyx
index 1e78c3837d9..007c2314fbf 100644
--- a/src/sage/graphs/cliquer.pyx
+++ b/src/sage/graphs/cliquer.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 r"""
 Interface with Cliquer (clique-related problems)
 
@@ -24,7 +25,6 @@ Methods
 -------
 """
 
-
 #*****************************************************************************
 #       Copyright (C) 2009 Nathann Cohen <nathann.cohen@gmail.com>
 #
@@ -89,7 +89,7 @@ def max_clique(graph):
 
     sig_free(list)
     graph_free(g)
-    return list_composition(b,d_inv)
+    return [d_inv[k] for k in b]
 
 
 # computes all the maximum clique of a graph and return its list
@@ -162,7 +162,7 @@ def all_max_clique(graph):
         if(list[i]!=-1):
             c.append(list[i])
         else:
-            b.append(list_composition(c,d_inv))
+            b.append([d_inv[k] for k in c])
             c=[]
 
     sig_free(list)
@@ -214,22 +214,3 @@ def clique_number(graph):
     graph_free(g)
     sig_off()
     return c
-
-
-def list_composition(a,b):
-    """
-    Composes a list ``a`` with a map ``b``.
-
-    EXAMPLES::
-
-        sage: from sage.graphs.cliquer import list_composition
-        sage: list_composition([1,3,'a'], {'a':'b', 1:2, 2:3, 3:4})
-        [2, 4, 'b']
-
-    """
-    value=[]
-    for i in a:
-        value.append(b[i])
-    return value
-
-
diff --git a/src/sage/graphs/comparability.pyx b/src/sage/graphs/comparability.pyx
index 5d7ec382f85..a32c79c4413 100644
--- a/src/sage/graphs/comparability.pyx
+++ b/src/sage/graphs/comparability.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 r"""
 Comparability and permutation graphs
 
@@ -209,6 +210,7 @@ Methods
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+
 from __future__ import print_function
 
 from cysignals.memory cimport sig_free
@@ -457,8 +459,8 @@ def is_comparability_MILP(g, certificate = False):
                     p.add_constraint(o[u,v] + o[vv,u] - o[vv,v] <= 1)
                     p.add_constraint(o[u,vv] + o[v,u] - o[v,vv] <= 1)
 
-                # If there is no edge there there are only two
-                # orientation possible (see the module's documentation
+                # If there is no edge, there are only two
+                # orientations possible (see the module's documentation
                 # about edges which imply each other)
                 else:
                     p.add_constraint(o[u,v] + o[vv,u] <= 1)
diff --git a/src/sage/graphs/digraph.py b/src/sage/graphs/digraph.py
index 269e7d5e095..936ba1cfd9a 100644
--- a/src/sage/graphs/digraph.py
+++ b/src/sage/graphs/digraph.py
@@ -70,6 +70,8 @@
     :meth:`~DiGraph.strongly_connected_components_subgraphs` | Returns the strongly connected components as a list of subgraphs.
     :meth:`~DiGraph.strongly_connected_component_containing_vertex` | Returns the strongly connected component containing a given vertex
     :meth:`~DiGraph.strongly_connected_components` | Returns the list of strongly connected components.
+    :meth:`~DiGraph.immediate_dominators` | Return the immediate dominators of all vertices reachable from `root`.
+    :meth:`~DiGraph.strong_articulation_points` | Return the strong articulation points of this digraph.
 
 
 **Acyclicity:**
@@ -3220,6 +3222,263 @@ def is_strongly_connected(self):
         except AttributeError:
             return len(self.strongly_connected_components()) == 1
 
+
+    def immediate_dominators(self, r, reverse=False):
+        r"""
+        Return the immediate dominators of all vertices reachable from `r`.
+
+        A flowgraph `G = (V, A, r)` is a digraph where every vertex in `V` is
+        reachable from a distinguished root vertex `r\in V`. In such digraph, a
+        vertex `w` dominates a vertex `v` if every path from `r` to `v` includes
+        `w`. Let `dom(v)` be the set of the vertices that dominate `v`.
+        Obviously, `r` and `v`, the trivial dominators of `v`, are in
+        `dom(v)`. For `v \neq r`, the immediate dominator of `v`, denoted by
+        `d(v)`, is the unique vertex `w \neq v` that dominates `v` and is
+        dominated by all the vertices in `dom(v)\setminus\{v\}`. The (immediate)
+        dominator tree is a directed tree (or arborescence) rooted at `r` that
+        is formed by the arcs `\{ (d(v), v)\mid v\in V\setminus\{r\}\}`.  See
+        [Ge2005]_ for more details.
+
+        This method implements the algorithm proposed in [CHK2001]_ which
+        performs very well in practice, although its worst case time complexity
+        is in `O(n^2)`.
+
+        INPUT:
+
+        - ``r`` -- a vertex of the digraph, the root of the immediate dominators
+          tree
+
+        - ``reverse`` -- boolean (default: ``False``); When set to ``True``, we
+          consider the reversed digraph in which out-neighbors become the
+          in-neighbors and vice-versa. This option is available only if the
+          backend of the digraph is :mod:`~SparseGraphBackend`.
+
+        OUTPUT: The (immediate) dominator tree rooted at `r`, encoded as a
+        predecessor dictionary.
+
+        EXAMPLES:
+
+        The output encodes a tree rooted at `r`::
+
+            sage: D = digraphs.Complete(4) * 2
+            sage: D.add_edges([(0, 4), (7, 3)])
+            sage: d = D.immediate_dominators(0)
+            sage: T = DiGraph([(d[u], u) for u in d if u != d[u]])
+            sage: Graph(T).is_tree()
+            True
+            sage: all(T.in_degree(u) <= 1 for u in T)
+            True
+
+        In a strongly connected digraph, the result depends on the root::
+
+            sage: D = digraphs.Circuit(5)
+            sage: D.immediate_dominators(0)
+            {0: 0, 1: 0, 2: 1, 3: 2, 4: 3}
+            sage: D.immediate_dominators(1)
+            {0: 4, 1: 1, 2: 1, 3: 2, 4: 3}
+
+        The (immediate) dominator tree contains only reachable vertices::
+
+            sage: P = digraphs.Path(5)
+            sage: P.immediate_dominators(0)
+            {0: 0, 1: 0, 2: 1, 3: 2, 4: 3}
+            sage: P.immediate_dominators(3)
+            {3: 3, 4: 3}
+
+        Immediate dominators in the reverse digraph::
+
+            sage: D = digraphs.Complete(5)+digraphs.Complete(4)
+            sage: D.add_edges([(0, 5), (1, 6), (7, 2)])
+            sage: idom = D.immediate_dominators(0, reverse=True)
+            sage: idom
+            {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: 7, 6: 7, 7: 2, 8: 7}
+            sage: D_reverse = D.reverse()
+            sage: D_reverse.immediate_dominators(0) == idom
+            True
+
+        .. SEEALSO::
+
+            - :wikipedia:`Dominator_(graph_theory)`
+            - :meth:`~DiGraph.strong_articulation_points`
+            - :meth:`~DiGraph.strongly_connected_components`
+
+        TESTS:
+
+        When `r` is not in the digraph::
+
+            sage: DiGraph().immediate_dominators(0)
+            Traceback (most recent call last):
+            ...
+            ValueError: the given root must be in the digraph
+
+        The reverse option is available only when the backend of the digraph is
+        :mod:`~SparseGraphBackend`::
+
+            sage: H = DiGraph(D.edges(), data_structure='static_sparse')
+            sage: H.immediate_dominators(0, reverse=True)
+            Traceback (most recent call last):
+            ...
+            ValueError: the reverse option is not available for this digraph
+
+        Comparison with the NetworkX method::
+
+            sage: import networkx
+            sage: D = digraphs.RandomDirectedGNP(20,0.1)
+            sage: d = D.immediate_dominators(0)
+            sage: dx = networkx.immediate_dominators(D.networkx_graph(), 0)
+            sage: all(d[i] == dx[i] for i in d) and all(d[i] == dx[i] for i in dx)
+            True
+        """
+        if r not in self:
+            raise ValueError("the given root must be in the digraph")
+
+        idom = {r: r}
+        n = self.order()
+        if reverse:
+            from sage.graphs.base.sparse_graph import SparseGraphBackend
+            if isinstance(self._backend, SparseGraphBackend):
+                pre_order = list(self._backend.depth_first_search(r, reverse=True))
+                number = {u: n-i for i, u in enumerate(pre_order)}
+                neighbor_iterator = self.neighbor_out_iterator
+            else:
+                raise ValueError("the reverse option is not available for this digraph")
+        else:
+            pre_order = list(self.depth_first_search(r))
+            number = {u: n-i for i, u in enumerate(pre_order)}
+            neighbor_iterator = self.neighbor_in_iterator
+        pre_order.pop(0)
+
+        def intersect(u, v):
+            while u != v:
+                while number[u] < number[v]:
+                    u = idom[u]
+                while number[u] > number[v]:
+                    v = idom[v]
+            return u
+
+        changed = True
+        while changed:
+            changed = False
+            for u in pre_order:
+                pred = [v for v in neighbor_iterator(u) if v in idom]
+                if not pred:
+                    continue
+                else:
+                    new_idom = pred[0]
+                    for v in pred[1:]:
+                        new_idom = intersect(new_idom, v)
+                if not u in idom or idom[u] != new_idom:
+                    idom[u] = new_idom
+                    changed = True
+
+        return idom
+
+    def strong_articulation_points(self):
+        r"""
+        Return the strong articulation points of this digraph.
+
+        A vertex is a strong articulation point if its deletion increases the
+        number of strongly connected components. This method implements the
+        algorithm described in [ILS2012]_. The time complexity is dominated by
+        the time complexity of the immediate dominators finding algorithm.
+
+        OUTPUT: The list of strong articulation points.
+
+        EXAMPLES:
+
+        Two cliques sharing a vertex::
+
+            sage: D = digraphs.Complete(4)
+            sage: D.add_clique([3, 4, 5, 6])
+            sage: D.strong_articulation_points()
+            [3]
+
+        Two cliques connected by some arcs::
+
+            sage: D = digraphs.Complete(4) * 2
+            sage: D.add_edges([(0, 4), (7, 3)])
+            sage: sorted( D.strong_articulation_points() )
+            [0, 3, 4, 7]
+            sage: D.add_edge(1, 5)
+            sage: sorted( D.strong_articulation_points() )
+            [3, 7]
+            sage: D.add_edge(6, 2)
+            sage: D.strong_articulation_points()
+            []
+
+        .. SEEALSO::
+
+            - :meth:`~DiGraph.strongly_connected_components`
+            - :meth:`~DiGraph.immediate_dominators`
+
+        TESTS:
+
+        All strong articulation points are found::
+
+            sage: def sap_naive(G):
+            ....:     nscc = len(G.strongly_connected_components())
+            ....:     S = []
+            ....:     for u in G:
+            ....:         H = copy(G)
+            ....:         H.delete_vertex(u)
+            ....:         if len(H.strongly_connected_components()) > nscc:
+            ....:             S.append(u)
+            ....:     return S
+            sage: D = digraphs.RandomDirectedGNP(20, 0.1)
+            sage: X = sap_naive(D)
+            sage: SAP = D.strong_articulation_points()
+            sage: set(X) == set(SAP)
+            True
+
+        Trivial cases::
+
+            sage: DiGraph().strong_articulation_points()
+            []
+            sage: DiGraph(1).strong_articulation_points()
+            []
+            sage: DiGraph(2).strong_articulation_points()
+            []
+        """
+        # The method is applied on each strongly connected component
+        if self.is_strongly_connected():
+            # Make a mutable copy of self
+            L = [ DiGraph( [(u, v) for u, v in self.edge_iterator(labels=0) if u != v],
+                               data_structure='sparse', immutable=False) ]
+        else:
+            # Get the list of strongly connected components of self as mutable
+            # subgraphs
+            L = [ self.subgraph(scc, immutable=False) for scc in self.strongly_connected_components() ]
+
+        SAP = []
+        for g in L:
+            n = g.order()
+            if n <= 1:
+                continue
+            if n == 2:
+                SAP.extend( g.vertices() )
+                continue
+
+            # 1. Choose arbitrarily a vertex r, and test whether r is a strong
+            # articulation point.
+            r = next(g.vertex_iterator())
+            E = g.incoming_edges(r) + g.outgoing_edges(r)
+            g.delete_vertex(r)
+            if not g.is_strongly_connected():
+                SAP.append(r)
+            g.add_edges(E)
+
+            # 2. Compute the set of non-trivial immediate dominators in g
+            Dr = set( g.immediate_dominators(r).values() )
+
+            # 3. Compute the set of non-trivial immediate dominators in the
+            # reverse digraph
+            DRr = set( g.immediate_dominators(r, reverse=True).values() )
+
+            # 4. Store D(r) + DR(r) - r
+            SAP.extend( Dr.union(DRr).difference([r]) )
+
+        return SAP
+
     def is_aperiodic(self):
         r"""
         Return whether the current ``DiGraph`` is aperiodic.
@@ -3510,10 +3769,6 @@ def flow_polytope(self, edges=None, ends=None):
 
         return Polyhedron(ieqs=ineqs, eqns=eqs)
 
-import types
-
-import sage.graphs.comparability
-DiGraph.is_transitive = types.MethodType(sage.graphs.comparability.is_transitive, None, DiGraph)
-
-from sage.graphs.base.static_sparse_graph import tarjan_strongly_connected_components
-DiGraph.strongly_connected_components = types.MethodType(tarjan_strongly_connected_components, None, DiGraph)
+    # Aliases to functions defined in other modules
+    from sage.graphs.comparability import is_transitive
+    from sage.graphs.base.static_sparse_graph import tarjan_strongly_connected_components as strongly_connected_components
diff --git a/src/sage/graphs/distances_all_pairs.pyx b/src/sage/graphs/distances_all_pairs.pyx
index 5965dd18fa0..955c08bc89a 100644
--- a/src/sage/graphs/distances_all_pairs.pyx
+++ b/src/sage/graphs/distances_all_pairs.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 r"""
 Distances/shortest paths between all pairs of vertices
 
@@ -833,7 +834,7 @@ def eccentricity(G, algorithm="standard"):
         sage: eccentricity(g)
         [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
 
-    TEST:
+    TESTS:
 
     All algorithms are valid::
 
@@ -1337,7 +1338,7 @@ def diameter(G, algorithm='iFUB', source=None):
         sage: lbm = diameter(G, algorithm='multi-sweep')
         sage: if not (lb2<=lbm and lbm<=d3): print("Something goes wrong!")
 
-    TEST:
+    TESTS:
 
     This was causing a segfault. Fixed in :trac:`17873` ::
 
diff --git a/src/sage/graphs/generators/classical_geometries.py b/src/sage/graphs/generators/classical_geometries.py
index dd9f75475ae..8a3f67170ce 100644
--- a/src/sage/graphs/generators/classical_geometries.py
+++ b/src/sage/graphs/generators/classical_geometries.py
@@ -212,7 +212,7 @@ def _orthogonal_polar_graph(m, q, sign="+", point_type=[0]):
     r"""
     A helper function to build ``OrthogonalPolarGraph`` and ``NO2,3,5`` graphs.
 
-    See see the `page of
+    See the `page of
     Andries Brouwer's website <http://www.win.tue.nl/~aeb/graphs/srghub.html>`_.
 
     INPUT:
@@ -333,7 +333,7 @@ def OrthogonalPolarGraph(m, q, sign="+"):
     r"""
     Returns the Orthogonal Polar Graph `O^{\epsilon}(m,q)`.
 
-    For more information on Orthogonal Polar graphs, see see the `page of
+    For more information on Orthogonal Polar graphs, see the `page of
     Andries Brouwer's website <http://www.win.tue.nl/~aeb/graphs/srghub.html>`_.
 
     INPUT:
diff --git a/src/sage/graphs/generators/intersection.py b/src/sage/graphs/generators/intersection.py
index 69d5a0c27e9..a81c4a641fb 100644
--- a/src/sage/graphs/generators/intersection.py
+++ b/src/sage/graphs/generators/intersection.py
@@ -333,7 +333,7 @@ def ToleranceGraph(tolrep):
         sage: g.is_isomorphic(graphs.PathGraph(3))
         True
 
-    TEST:
+    TESTS:
 
     Giving negative third value::
 
diff --git a/src/sage/graphs/generic_graph.py b/src/sage/graphs/generic_graph.py
index 8825adf468f..cecdcfa27ff 100644
--- a/src/sage/graphs/generic_graph.py
+++ b/src/sage/graphs/generic_graph.py
@@ -303,7 +303,7 @@
     :meth:`~GenericGraph.disjoint_routed_paths` | Return a set of disjoint routed paths.
     :meth:`~GenericGraph.dominating_set` | Return a minimum dominating set of the graph
     :meth:`~GenericGraph.subgraph_search` | Return a copy of ``G`` in ``self``.
-    :meth:`~GenericGraph.subgraph_search_count` | Return the number of labelled occurences of ``G`` in ``self``.
+    :meth:`~GenericGraph.subgraph_search_count` | Return the number of labelled occurrences of ``G`` in ``self``.
     :meth:`~GenericGraph.subgraph_search_iterator` | Return an iterator over the labelled copies of ``G`` in ``self``.
     :meth:`~GenericGraph.characteristic_polynomial` | Return the characteristic polynomial of the adjacency matrix of the (di)graph.
     :meth:`~GenericGraph.genus` | Return the minimal genus of the graph.
@@ -1483,8 +1483,7 @@ def igraph_graph(self, vertex_attrs={}, edge_attrs={}):
         try:
             import igraph
         except ImportError:
-            raise ImportError("The package igraph is not available. To " +
-                              "install it, run 'sage -i python_igraph'")
+            raise PackageNotFoundError("python_igraph")
 
         v_to_int = {v: i for i, v in enumerate(self.vertices())}
         edges = [(v_to_int[v], v_to_int[w]) for v,w in self.edge_iterator(labels=False)]
@@ -8078,7 +8077,7 @@ def flow(self, x, y, value_only=True, integer=False, use_edge_labels=True, verte
            sage: abs(flow_ff-flow_igraph) < 0.00001         # optional python_igraph
            True
         """
-        from sage.misc.package import is_package_installed
+        from sage.misc.package import is_package_installed, PackageNotFoundError
         self._scream_if_not_simple(allow_loops=True)
         if vertex_bound and algorithm in ["FF", "igraph"]:
             raise ValueError("This method does not support both " +
@@ -8108,8 +8107,7 @@ def flow(self, x, y, value_only=True, integer=False, use_edge_labels=True, verte
             try:
                 import igraph
             except ImportError:
-                raise ImportError("The igraph library is not available. " +
-                                 "Please, install it with sage -i python_igraph")
+                raise PackageNotFoundError("python_igraph")
             vertices = self.vertices()
             x_int = vertices.index(x)
             y_int = vertices.index(y)
@@ -12147,7 +12145,7 @@ def subgraph_search(self, G, induced=False):
 
     def subgraph_search_count(self, G, induced=False):
         r"""
-        Returns the number of labelled occurences of ``G`` in ``self``.
+        Return the number of labelled occurrences of ``G`` in ``self``.
 
         INPUT:
 
@@ -16435,7 +16433,7 @@ def average_distance(self, by_weight=False, algorithm=None,
            results in the theory of the Wiener number. *Indian Journal of
            Chemistry*, 32A:651--661, 1993.
 
-        TEST:
+        TESTS:
 
         Giving an empty graph::
 
@@ -20782,12 +20780,6 @@ def automorphism_group(self, partition=None, verbosity=0,
           "sage" uses Sage's implementation. If set to ``None`` (default), bliss
           is used when available.
 
-        .. WARNING::
-
-            Since :trac:`14319` the domain of the automorphism group is equal to
-            the graph's vertex set, and the ``translation`` argument has become
-            useless.
-
         OUTPUT: The order of the output is group, order, orbits. However, there
         are options to turn each of these on or off.
 
@@ -21874,20 +21866,12 @@ def is_cayley(self, return_group = False, mapping = False,
         else:
             return c
 
-import types
-
-import sage.graphs.distances_all_pairs
-GenericGraph.distances_distribution = types.MethodType(sage.graphs.distances_all_pairs.distances_distribution, None, GenericGraph)
-
-from sage.graphs.base.boost_graph import dominator_tree
-GenericGraph.dominator_tree = types.MethodType(dominator_tree, None, GenericGraph)
-
-from sage.graphs.base.static_sparse_graph import spectral_radius
-GenericGraph.spectral_radius = types.MethodType(spectral_radius, None, GenericGraph)
+    # Aliases to functions defined in other modules
+    from sage.graphs.distances_all_pairs import distances_distribution
+    from sage.graphs.base.boost_graph import dominator_tree
+    from sage.graphs.base.static_sparse_graph import spectral_radius
+    from sage.graphs.line_graph import line_graph
 
-# From Python modules
-import sage.graphs.line_graph
-GenericGraph.line_graph = sage.graphs.line_graph.line_graph
 
 def tachyon_vertex_plot(g, bgcolor=(1,1,1),
                         vertex_colors=None,
diff --git a/src/sage/graphs/graph.py b/src/sage/graphs/graph.py
index d92d8bf8c02..4e35ec77389 100644
--- a/src/sage/graphs/graph.py
+++ b/src/sage/graphs/graph.py
@@ -1360,7 +1360,7 @@ def sparse6_string(self):
             sage: Graph(':?',data_structure="sparse") == G
             True
 
-        TEST:
+        TESTS:
 
         Check that :trac:`18445` is fixed::
 
@@ -2818,7 +2818,8 @@ def treewidth(self,k=None,certificate=False,algorithm=None):
 
         if k is not None and k >= g.order()-1:
             if certificate:
-                return Graph({sage.sets.set.Set(g.vertices()):[]},
+                from sage.sets.set import Set
+                return Graph({Set(g.vertices()):[]},
                              name="Tree decomposition")
             return True
 
@@ -3010,7 +3011,7 @@ def is_perfect(self, certificate = False):
             sage: g.is_perfect(certificate = True)
             Subgraph of (Petersen graph): Graph on 5 vertices
 
-        TEST:
+        TESTS:
 
         Check that :trac:`13546` has been fixed::
 
@@ -6006,7 +6007,8 @@ def clique_maximum(self,  algorithm="Cliquer"):
             try:
                 from sage.graphs.mcqd import mcqd
             except ImportError:
-                raise ImportError("Please install the mcqd package")
+                from sage.misc.package import PackageNotFoundError
+                raise PackageNotFoundError("mcqd")
             return mcqd(self)
         else:
             raise NotImplementedError("Only 'MILP', 'Cliquer' and 'mcqd' are supported.")
@@ -6097,7 +6099,8 @@ def clique_number(self, algorithm="Cliquer", cliques=None):
             try:
                 from sage.graphs.mcqd import mcqd
             except ImportError:
-                raise ImportError("Please install the mcqd package")
+                from sage.misc.package import PackageNotFoundError
+                raise PackageNotFoundError("mcqd")
             return len(mcqd(self))
         else:
             raise NotImplementedError("Only 'networkx' 'MILP' 'Cliquer' and 'mcqd' are supported.")
@@ -7728,7 +7731,7 @@ def has_perfect_matching(self, algorithm="Edmonds", solver=None, verbose=0):
             find a matching of maximal cardinality, then check whether this
             cardinality is half the number of vertices of the graph.
 
-          - ``"LP_matching"`` uses uses a Linear Program to find a matching of
+          - ``"LP_matching"`` uses a Linear Program to find a matching of
             maximal cardinality, then check whether this cardinality is half the
             number of vertices of the graph.
 
@@ -7817,60 +7820,24 @@ def has_perfect_matching(self, algorithm="Edmonds", solver=None, verbose=0):
         else:
             raise ValueError('algorithm must be set to "Edmonds", "LP_matching" or "LP"')
 
+    # Aliases to functions defined in other modules
+    from sage.graphs.weakly_chordal import is_long_hole_free, is_long_antihole_free, is_weakly_chordal
+    from sage.graphs.asteroidal_triples import is_asteroidal_triple_free
+    from sage.graphs.chrompoly import chromatic_polynomial
+    from sage.graphs.graph_decompositions.rankwidth import rank_decomposition
+    from sage.graphs.graph_decompositions.vertex_separation import pathwidth
+    from sage.graphs.matchpoly import matching_polynomial
+    from sage.graphs.cliquer import all_max_clique as cliques_maximum
+    from sage.graphs.spanning_tree import random_spanning_tree
+    from sage.graphs.graph_decompositions.graph_products import is_cartesian_product
+    from sage.graphs.distances_all_pairs import is_distance_regular
+    from sage.graphs.base.static_dense_graph import is_strongly_regular
+    from sage.graphs.line_graph import is_line_graph
+    from sage.graphs.tutte_polynomial import tutte_polynomial
+    from sage.graphs.lovasz_theta import lovasz_theta
+    from sage.graphs.partial_cube import is_partial_cube
+    from sage.graphs.orientations import strong_orientations_iterator
 
-# Aliases to functions defined in Cython modules
-import types
-
-import sage.graphs.weakly_chordal
-Graph.is_long_hole_free         = types.MethodType(sage.graphs.weakly_chordal.is_long_hole_free, None, Graph)
-Graph.is_long_antihole_free     = types.MethodType(sage.graphs.weakly_chordal.is_long_antihole_free, None, Graph)
-Graph.is_weakly_chordal         = types.MethodType(sage.graphs.weakly_chordal.is_weakly_chordal, None, Graph)
-
-import sage.graphs.asteroidal_triples
-Graph.is_asteroidal_triple_free = types.MethodType(sage.graphs.asteroidal_triples.is_asteroidal_triple_free, None, Graph)
-
-import sage.graphs.chrompoly
-Graph.chromatic_polynomial      = types.MethodType(sage.graphs.chrompoly.chromatic_polynomial, None, Graph)
-
-import sage.graphs.graph_decompositions.rankwidth
-Graph.rank_decomposition        = types.MethodType(sage.graphs.graph_decompositions.rankwidth.rank_decomposition, None, Graph)
-
-import sage.graphs.graph_decompositions.vertex_separation
-Graph.pathwidth                 = types.MethodType(sage.graphs.graph_decompositions.vertex_separation.pathwidth, None, Graph)
-
-import sage.graphs.matchpoly
-Graph.matching_polynomial       = types.MethodType(sage.graphs.matchpoly.matching_polynomial, None, Graph)
-
-import sage.graphs.cliquer
-Graph.cliques_maximum           = types.MethodType(sage.graphs.cliquer.all_max_clique, None, Graph)
-
-import sage.graphs.spanning_tree
-Graph.random_spanning_tree      = types.MethodType(sage.graphs.spanning_tree.random_spanning_tree, None, Graph)
-
-import sage.graphs.graph_decompositions.graph_products
-Graph.is_cartesian_product      = types.MethodType(sage.graphs.graph_decompositions.graph_products.is_cartesian_product, None, Graph)
-
-import sage.graphs.distances_all_pairs
-Graph.is_distance_regular       = types.MethodType(sage.graphs.distances_all_pairs.is_distance_regular, None, Graph)
-
-import sage.graphs.base.static_dense_graph
-Graph.is_strongly_regular       = types.MethodType(sage.graphs.base.static_dense_graph.is_strongly_regular, None, Graph)
-
-# From Python modules
-import sage.graphs.line_graph
-Graph.is_line_graph             = sage.graphs.line_graph.is_line_graph
-
-from sage.graphs.tutte_polynomial import tutte_polynomial
-Graph.tutte_polynomial          = tutte_polynomial
-
-from sage.graphs.lovasz_theta import lovasz_theta
-Graph.lovasz_theta              = lovasz_theta
-
-from sage.graphs.partial_cube import is_partial_cube
-Graph.is_partial_cube           = is_partial_cube
-
-from sage.graphs.orientations import strong_orientations_iterator
-Graph.strong_orientations_iterator    =    strong_orientations_iterator
 
 _additional_categories = {
     Graph.is_long_hole_free         : "Graph properties",
diff --git a/src/sage/graphs/graph_decompositions/graph_products.pyx b/src/sage/graphs/graph_decompositions/graph_products.pyx
index f290a8f91ba..97ee3166f0e 100644
--- a/src/sage/graphs/graph_decompositions/graph_products.pyx
+++ b/src/sage/graphs/graph_decompositions/graph_products.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 r"""
 Products of graphs
 
diff --git a/src/sage/graphs/graph_decompositions/rankwidth.pyx b/src/sage/graphs/graph_decompositions/rankwidth.pyx
index 26871baad48..768bab26742 100644
--- a/src/sage/graphs/graph_decompositions/rankwidth.pyx
+++ b/src/sage/graphs/graph_decompositions/rankwidth.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 r"""
 Rank Decompositions of graphs
 
diff --git a/src/sage/graphs/graph_decompositions/vertex_separation.pyx b/src/sage/graphs/graph_decompositions/vertex_separation.pyx
index 4683b242a17..02969d7285d 100644
--- a/src/sage/graphs/graph_decompositions/vertex_separation.pyx
+++ b/src/sage/graphs/graph_decompositions/vertex_separation.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 r"""
 Vertex separation
 
@@ -324,7 +325,7 @@ def lower_bound(G):
         sage: lower_bound(g)
         1
 
-    TEST:
+    TESTS:
 
     Given anything else than a Graph or a DiGraph::
 
@@ -669,7 +670,7 @@ def path_decomposition(G, algorithm = "BAB", cut_off=None, upper_bound=None, ver
         sage: pw, L = path_decomposition(g, algorithm = "MILP"); pw
         2
 
-    TEST:
+    TESTS:
 
     Given anything else than a Graph::
 
@@ -918,7 +919,7 @@ def vertex_separation_exp(G, verbose = False):
         sage: vertex_separation_exp(g)
         (1, [0, 1, 2, 3, 4, 5])
 
-    TEST:
+    TESTS:
 
     Given anything else than a Graph or a DiGraph::
 
@@ -1105,7 +1106,7 @@ def is_valid_ordering(G, L):
         sage: vertex_separation.is_valid_ordering(G, [1,2])
         False
 
-    TEST:
+    TESTS:
 
     Giving anything else than a Graph or a DiGraph::
 
diff --git a/src/sage/graphs/graph_generators.py b/src/sage/graphs/graph_generators.py
index 3cb6e4c879b..76678fe8487 100644
--- a/src/sage/graphs/graph_generators.py
+++ b/src/sage/graphs/graph_generators.py
@@ -1205,9 +1205,9 @@ def fullerenes(self, order, ipr=False):
         .. [buckygen] \G. Brinkmann, J. Goedgebeur and B.D. McKay, Generation of Fullerenes,
           Journal of Chemical Information and Modeling, 52(11):2910-2918, 2012.
         """
-        from sage.misc.package import is_package_installed
+        from sage.misc.package import is_package_installed, PackageNotFoundError
         if not is_package_installed("buckygen"):
-            raise TypeError("the optional buckygen package is not installed")
+            raise PackageNotFoundError("buckygen")
 
         # number of vertices should be positive
         if order < 0:
@@ -1290,9 +1290,9 @@ def fusenes(self, hexagon_count, benzenoids=False):
         .. [benzene] \G. Brinkmann, G. Caporossi and P. Hansen, A Constructive Enumeration of Fusenes and Benzenoids,
           Journal of Algorithms, 45:155-166, 2002.
         """
-        from sage.misc.package import is_package_installed
+        from sage.misc.package import is_package_installed, PackageNotFoundError
         if not is_package_installed("benzene"):
-            raise TypeError("the optional benzene package is not installed")
+            raise PackageNotFoundError("benzene")
 
         # number of hexagons should be positive
         if hexagon_count < 0:
@@ -1443,9 +1443,9 @@ def planar_graphs(self, order, minimum_degree=None,
         .. [plantri] \G. Brinkmann and B.D. McKay, Fast generation of planar graphs,
            MATCH-Communications in Mathematical and in Computer Chemistry, 58(2):323-357, 2007.
         """
-        from sage.misc.package import is_package_installed
+        from sage.misc.package import is_package_installed, PackageNotFoundError
         if not is_package_installed("plantri"):
-            raise TypeError("the optional plantri package is not installed")
+            raise PackageNotFoundError("plantri")
 
         # number of vertices should be positive
         if order < 0:
@@ -1642,9 +1642,9 @@ def triangulations(self, order, minimum_degree=None, minimum_connectivity=None,
             sage: [g.size() for g in graphs.triangulations(6, minimum_connectivity=3)] # optional plantri
             [12, 12]
         """
-        from sage.misc.package import is_package_installed
+        from sage.misc.package import is_package_installed, PackageNotFoundError
         if not is_package_installed("plantri"):
-            raise TypeError("the optional plantri package is not installed")
+            raise PackageNotFoundError("plantri")
 
         # number of vertices should be positive
         if order < 0:
@@ -1796,9 +1796,9 @@ def quadrangulations(self, order, minimum_degree=None, minimum_connectivity=None
             sage: [len(g) for g in graphs.quadrangulations(12, no_nonfacial_quadrangles=True, dual=True)]  # optional plantri
             [10, 10]
         """
-        from sage.misc.package import is_package_installed
+        from sage.misc.package import is_package_installed, PackageNotFoundError
         if not is_package_installed("plantri"):
-            raise TypeError("the optional plantri package is not installed")
+            raise PackageNotFoundError("plantri")
 
         # number of vertices should be positive
         if order < 0:
diff --git a/src/sage/graphs/graph_latex.py b/src/sage/graphs/graph_latex.py
index d3b85f95f8a..dd94b514009 100644
--- a/src/sage/graphs/graph_latex.py
+++ b/src/sage/graphs/graph_latex.py
@@ -222,7 +222,7 @@
     \end{tikzpicture}
 
 
-TEST:
+TESTS:
 
 This graph will look horrible, but it illustrates (and tests) a
 great variety of the possible options available through Sage's
diff --git a/src/sage/graphs/independent_sets.pyx b/src/sage/graphs/independent_sets.pyx
index 9ec5540f53a..dcd6dd19451 100644
--- a/src/sage/graphs/independent_sets.pyx
+++ b/src/sage/graphs/independent_sets.pyx
@@ -103,7 +103,7 @@ cdef class IndependentSets:
         sage: number_of
         [1, 10, 30, 30, 5, 0, 0, 0, 0, 0]
 
-    It is also possible to define an an iterator over all independent sets of a
+    It is also possible to define an iterator over all independent sets of a
     given cardinality. Note, however, that Sage will generate them *all*, to
     return only those that satisfy the cardinality constraints. Getting the list
     of independent sets of size 4 in this way can thus take a very long time::
diff --git a/src/sage/graphs/lovasz_theta.py b/src/sage/graphs/lovasz_theta.py
index 39c6525bf1f..f150dca25f4 100644
--- a/src/sage/graphs/lovasz_theta.py
+++ b/src/sage/graphs/lovasz_theta.py
@@ -37,11 +37,11 @@ def lovasz_theta(graph):
 
     .. NOTE::
 
-        - Implemented for undirected graphs only. Use to_undirected to convert a
-          digraph to an undirected graph.
+        - Implemented for undirected graphs only. Use ``to_undirected``
+          to convert a digraph to an undirected graph.
 
         - This function requires the optional package ``csdp``, which you can
-          install with with ``sage -i csdp``.
+          install with ``sage -i csdp``.
 
     EXAMPLES::
 
diff --git a/src/sage/graphs/matchpoly.pyx b/src/sage/graphs/matchpoly.pyx
index 8c6b87fca3c..96e949c1fd7 100644
--- a/src/sage/graphs/matchpoly.pyx
+++ b/src/sage/graphs/matchpoly.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 """
 Matching Polynomial
 
diff --git a/src/sage/graphs/planarity.pyx b/src/sage/graphs/planarity.pyx
index 7c6cd34a0a3..6ef8c1278ca 100644
--- a/src/sage/graphs/planarity.pyx
+++ b/src/sage/graphs/planarity.pyx
@@ -76,10 +76,10 @@ def is_planar(g, kuratowski=False, set_pos=False, set_embedding=False, circular=
         True
 
     There were some problems with ``set_pos`` stability in the past,
-    so let's check if this this runs without exception::
+    so let's check if this runs without exception::
 
-        sage: for i,g in enumerate(atlas_graphs):                         # long time
-        ....:     if (not g.is_connected() or i==0):
+        sage: for i, g in enumerate(atlas_graphs):           # long time
+        ....:     if (not g.is_connected() or i == 0):
         ....:         continue
         ....:     _ = g.is_planar(set_embedding=True, set_pos=True)
     """
diff --git a/src/sage/graphs/spanning_tree.pyx b/src/sage/graphs/spanning_tree.pyx
index 690dcaaeed7..f332aac8f93 100644
--- a/src/sage/graphs/spanning_tree.pyx
+++ b/src/sage/graphs/spanning_tree.pyx
@@ -52,24 +52,20 @@ Methods
 -------
 """
 
-###########################################################################
-# Copyright (c) 2007 Jason Grout <jason-sage@creativetrax.com>
-# Copyright (c) 2009 Mike Hansen <mhansen@gmail.com>
-# Copyright (c) 2010 Gregory McWhirter <gmcwhirt@uci.edu>
-# Copyright (c) 2010 Minh Van Nguyen <nguyenminh2@gmail.com>
+#*****************************************************************************
+#       Copyright (c) 2007 Jason Grout <jason-sage@creativetrax.com>
+#       Copyright (c) 2009 Mike Hansen <mhansen@gmail.com>
+#       Copyright (c) 2010 Gregory McWhirter <gmcwhirt@uci.edu>
+#       Copyright (c) 2010 Minh Van Nguyen <nguyenminh2@gmail.com>
 #
-# This program is free software; you can redistribute it and/or modify
+# This program is free software: you can redistribute it and/or modify
 # it under the terms of the GNU General Public License as published by
-# the Free Software Foundation; either version 2 of the License, or
+# the Free Software Foundation, either version 2 of the License, or
 # (at your option) any later version.
-#
-# This program is distributed in the hope that it will be useful,
-# but WITHOUT ANY WARRANTY; without even the implied warranty of
-# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-# GNU General Public License for more details.
-#
-# http://www.gnu.org/licenses/
-###########################################################################
+#                  http://www.gnu.org/licenses/
+#*****************************************************************************
+
+cimport cython
 
 
 cpdef kruskal(G, wfunction=None, bint check=False):
@@ -340,6 +336,7 @@ cpdef kruskal(G, wfunction=None, bint check=False):
     return sorted(T)
 
 
+@cython.binding(True)
 def random_spanning_tree(self, output_as_graph=False):
     r"""
     Return a random spanning tree of the graph.
diff --git a/src/sage/graphs/strongly_regular_db.pyx b/src/sage/graphs/strongly_regular_db.pyx
index f99d9b2cca0..88beb17837b 100644
--- a/src/sage/graphs/strongly_regular_db.pyx
+++ b/src/sage/graphs/strongly_regular_db.pyx
@@ -2251,7 +2251,7 @@ def SRG_276_140_58_84():
     r"""
     Return a `(276, 140, 58, 84)`-strongly regular graph.
 
-    The graph is built from from
+    The graph is built from
     :meth:`~sage.graphs.graph_generators.GraphGenerators.McLaughlinGraph`, with
     an added isolated vertex. We then perform a
     :meth:`~Graph.seidel_switching` on a set of 28 disjoint 5-cliques, which
diff --git a/src/sage/graphs/tutte_polynomial.py b/src/sage/graphs/tutte_polynomial.py
index 9df75d740ed..a67e3ab5747 100644
--- a/src/sage/graphs/tutte_polynomial.py
+++ b/src/sage/graphs/tutte_polynomial.py
@@ -41,7 +41,6 @@
 
 from contextlib import contextmanager
 from sage.misc.lazy_attribute import lazy_attribute
-from sage.graphs.graph import Graph
 from sage.misc.misc_c import prod
 from sage.rings.integer_ring import ZZ
 from sage.misc.decorators import sage_wraps
@@ -197,6 +196,7 @@ def underlying_graph(G):
         sage: underlying_graph(G).edges()
         [(0, 1, None)]
     """
+    from sage.graphs.graph import Graph
     g = Graph()
     g.allow_loops(True)
     for edge in set(G.edges(labels=False)):
diff --git a/src/sage/graphs/weakly_chordal.pyx b/src/sage/graphs/weakly_chordal.pyx
index f3b2faa1838..c8aad629b95 100644
--- a/src/sage/graphs/weakly_chordal.pyx
+++ b/src/sage/graphs/weakly_chordal.pyx
@@ -1,3 +1,4 @@
+# cython: binding=True
 r"""
 Weakly chordal graphs
 
diff --git a/src/sage/groups/braid.py b/src/sage/groups/braid.py
index fa2639ba424..2a9368fa62c 100644
--- a/src/sage/groups/braid.py
+++ b/src/sage/groups/braid.py
@@ -1676,7 +1676,7 @@ def _permutation_braid_Tietze(self, p):
             return ()
         pl = p
         l = []
-        while pl.length() > 0:
+        while pl.length():
             i = 1
             while i<max(pl):
                 if pl(i)>pl(i+1):
@@ -1684,7 +1684,7 @@ def _permutation_braid_Tietze(self, p):
                     pl = Permutation([(i, i+1)])*pl
                     i = 1
                 else:
-                    i = i+1
+                    i += 1
         return tuple(l)
 
     @cached_method
@@ -1696,7 +1696,8 @@ def _permutation_braid(self, p):
 
         - The braid induces the given permutation.
 
-        - The braid is positive (that is, it can be writen without using the inverses of the generators).
+        - The braid is positive (that is, it can be written without
+          using the inverses of the generators).
 
         - Every two strands cross each other at most once.
 
diff --git a/src/sage/groups/finitely_presented.py b/src/sage/groups/finitely_presented.py
index f2ebf57ac23..1c76552994f 100644
--- a/src/sage/groups/finitely_presented.py
+++ b/src/sage/groups/finitely_presented.py
@@ -118,17 +118,15 @@
 - Miguel Angel Marco Buzunariz
 """
 
-##############################################################################
+#*****************************************************************************
 #       Copyright (C) 2012 Miguel Angel Marco Buzunariz <mmarco@unizar.es>
 #
-#  Distributed under the terms of the GNU General Public License (GPL)
-#
-#  The full text of the GPL is available at:
-#
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
 #                  http://www.gnu.org/licenses/
-##############################################################################
-
-import types
+#*****************************************************************************
 
 from sage.groups.group import Group
 from sage.groups.libgap_wrapper import ParentLibGAP, ElementLibGAP
@@ -147,7 +145,6 @@
 from sage.rings.integer_ring import IntegerRing
 from sage.functions.generalized import sign
 from sage.matrix.constructor import matrix
-from sage.groups.generic import structure_description
 from sage.categories.morphism import SetMorphism
 
 
@@ -1042,7 +1039,7 @@ def direct_product(self, H, reduced=False, new_names=True):
           the group to be returned. If ``False``, the group to be returned
           keeps the generator names of the two groups forming the direct
           product. Note that one cannot ask to reduce the output and ask
-          to keep the old variable names, as they they may change meaning
+          to keep the old variable names, as they may change meaning
           in the output group if its presentation is reduced.
 
         OUTPUT:
@@ -1574,4 +1571,4 @@ def rewriting_system(self):
         """
         return RewritingSystem(self)
 
-FinitelyPresentedGroup.structure_description = types.MethodType(structure_description, None, FinitelyPresentedGroup)
+    from sage.groups.generic import structure_description
diff --git a/src/sage/groups/generic.py b/src/sage/groups/generic.py
index b970076ed5b..32bafe93dfd 100644
--- a/src/sage/groups/generic.py
+++ b/src/sage/groups/generic.py
@@ -1403,7 +1403,7 @@ def structure_description(G, latex=False):
         'A8'
     """
     import re
-    from sage.misc.package import is_package_installed
+    from sage.misc.package import is_package_installed, PackageNotFoundError
     def correct_dihedral_degree(match):
         return "%sD%d" % (match.group(1), int(match.group(2))/2)
 
@@ -1411,7 +1411,7 @@ def correct_dihedral_degree(match):
         description = str(G._gap_().StructureDescription())
     except RuntimeError:
         if not is_package_installed('database_gap'):
-            raise RuntimeError("You must install the optional database_gap package first.")
+            raise PackageNotFoundError("database_gap")
         raise
 
     description = re.sub(r"(\A|\W)D(\d+)", correct_dihedral_degree, description)
diff --git a/src/sage/groups/group.pyx b/src/sage/groups/group.pyx
index c6ad8181ae2..6aef02902f3 100644
--- a/src/sage/groups/group.pyx
+++ b/src/sage/groups/group.pyx
@@ -16,6 +16,7 @@ Base class for groups
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 import random
 
diff --git a/src/sage/groups/libgap_wrapper.pxd b/src/sage/groups/libgap_wrapper.pxd
index 5ab5dad1776..a7c689e201f 100644
--- a/src/sage/groups/libgap_wrapper.pxd
+++ b/src/sage/groups/libgap_wrapper.pxd
@@ -5,3 +5,4 @@ from sage.libs.gap.element cimport GapElement
 cdef class ElementLibGAP(MultiplicativeGroupElement):
     cdef GapElement _libgap
     cpdef GapElement gap(self)
+    cpdef _mul_(self, other)
diff --git a/src/sage/groups/matrix_gps/group_element.pxd b/src/sage/groups/matrix_gps/group_element.pxd
index 82355a1f7db..df5f99aeca9 100644
--- a/src/sage/groups/matrix_gps/group_element.pxd
+++ b/src/sage/groups/matrix_gps/group_element.pxd
@@ -7,6 +7,7 @@ cdef class MatrixGroupElement_generic(MultiplicativeGroupElement):
     cdef public Matrix _matrix
 
     cpdef _act_on_(self, x, bint self_on_left)
+    cpdef _mul_(self, other)
     cpdef list list(self)
 
 cdef class MatrixGroupElement_gap(ElementLibGAP):
diff --git a/src/sage/groups/matrix_gps/matrix_group.py b/src/sage/groups/matrix_gps/matrix_group.py
index 5e977e6d823..ddafa90daf7 100644
--- a/src/sage/groups/matrix_gps/matrix_group.py
+++ b/src/sage/groups/matrix_gps/matrix_group.py
@@ -36,19 +36,18 @@
 - Simon King (2010-05): Improve invariant_generators by using GAP
   for the construction of the Reynolds operator in Singular.
 """
-from __future__ import absolute_import
 
-##############################################################################
+#*****************************************************************************
 #       Copyright (C) 2006 David Joyner and William Stein <wstein@gmail.com>
 #
-#  Distributed under the terms of the GNU General Public License (GPL)
-#
-#  The full text of the GPL is available at:
-#
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
 #                  http://www.gnu.org/licenses/
-##############################################################################
+#*****************************************************************************
 
-import types
+from __future__ import absolute_import
 
 from sage.rings.integer import is_Integer
 from sage.rings.ring import is_Ring
@@ -59,7 +58,6 @@
 from sage.structure.richcmp import (richcmp_not_equal, rich_to_bool,
                                     richcmp_method, richcmp)
 from sage.misc.cachefunc import cached_method
-from sage.groups.generic import structure_description
 from sage.groups.group import Group
 from sage.groups.libgap_wrapper import ParentLibGAP
 from sage.groups.libgap_mixin import GroupMixinLibGAP
@@ -696,4 +694,4 @@ def _subgroup_constructor(self, libgap_subgroup):
         return FinitelyGeneratedMatrixGroup_gap(self.degree(), self.base_ring(),
                                                 libgap_subgroup, ambient=self)
 
-MatrixGroup_gap.structure_description = types.MethodType(structure_description, None, MatrixGroup_gap)
+    from sage.groups.generic import structure_description
diff --git a/src/sage/groups/old.pyx b/src/sage/groups/old.pyx
index 07c524f3fbb..bd38ab2f1d0 100644
--- a/src/sage/groups/old.pyx
+++ b/src/sage/groups/old.pyx
@@ -16,6 +16,7 @@ Base class for groups
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 doc="""
 Base class for all groups
diff --git a/src/sage/groups/perm_gps/partn_ref/canonical_augmentation.pyx b/src/sage/groups/perm_gps/partn_ref/canonical_augmentation.pyx
index 8b34a386bc3..8922e525efa 100644
--- a/src/sage/groups/perm_gps/partn_ref/canonical_augmentation.pyx
+++ b/src/sage/groups/perm_gps/partn_ref/canonical_augmentation.pyx
@@ -134,10 +134,10 @@ I. ``canonical_parent``:
 
     Apply the ``permutation`` to the ``child``, determine an arbitrary but fixed
     parent, apply the inverse of ``permutation`` to that parent, and return the
-    resulting object. Must also set the integer ``degree`` points to to the
+    resulting object. Must also set the integer ``degree`` points to the
     degree of the returned object.
 
-NOTE:
+.. NOTE::
 
     It is a good idea to try to implement an augmentation scheme where the
     degree of objects on each level of the augmentation tree is constant. The
diff --git a/src/sage/groups/perm_gps/permgroup.py b/src/sage/groups/perm_gps/permgroup.py
index 087899153bc..d8db0625de3 100644
--- a/src/sage/groups/perm_gps/permgroup.py
+++ b/src/sage/groups/perm_gps/permgroup.py
@@ -137,7 +137,6 @@
 
 from sage.misc.randstate import current_randstate
 import sage.groups.old as group
-import types
 
 from sage.rings.all import QQ, Integer
 from sage.interfaces.expect import is_ExpectElement
@@ -146,12 +145,11 @@
 from sage.groups.abelian_gps.abelian_group import AbelianGroup
 from sage.misc.cachefunc import cached_method
 from sage.groups.class_function import ClassFunction
-from sage.misc.package import is_package_installed
+from sage.misc.package import is_package_installed, PackageNotFoundError
 from sage.sets.finite_enumerated_set import FiniteEnumeratedSet
 from sage.categories.all import FiniteEnumeratedSets
 from sage.groups.conjugacy_classes import ConjugacyClassGAP
 from sage.functions.other import factorial
-from sage.groups.generic import structure_description
 from sage.structure.richcmp import (richcmp_method, richcmp_not_equal,
                                     richcmp, rich_to_bool, op_EQ)
 
@@ -195,7 +193,7 @@ def hap_decorator(f):
     @wraps(f)
     def wrapped(self, n, p=0):
         if not is_package_installed('gap_packages'):
-            raise RuntimeError("You must install the optional gap_packages package.")
+            raise PackageNotFoundError("gap_packages")
         load_hap()
         from sage.arith.all import is_prime
         if not (p == 0 or is_prime(p)):
@@ -1707,7 +1705,7 @@ def group_id(self):
             return [Integer(n) for n in self._gap_().IdGroup()]
         except RuntimeError:
             if not is_package_installed('database_gap'):
-                raise RuntimeError("You must install the optional database_gap package first.")
+                raise PackageNotFoundError("database_gap")
             raise
 
     def id(self):
@@ -1760,7 +1758,7 @@ def group_primitive_id(self):
             return Integer(self._gap_().PrimitiveIdentification())
         except RuntimeError:
             if not is_package_installed('database_gap'):
-                raise RuntimeError("You must install the optional database_gap package first.")
+                raise PackageNotFoundError("database_gap")
             raise
 
     def center(self):
@@ -4134,7 +4132,7 @@ def poincare_series(self, p=2, n=10):
 
         """
         if not is_package_installed('gap_packages'):
-            raise RuntimeError("You must install the optional gap_packages package.")
+            raise PackageNotFoundError("gap_packages")
         load_hap()
         from sage.arith.all import is_prime
         if not (p == 0 or is_prime(p)):
@@ -4193,7 +4191,8 @@ def upper_central_series(self):
         UCS = self._gap_().UpperCentralSeriesOfGroup()
         return [self.subgroup(gap_group=group) for group in UCS]
 
-PermutationGroup_generic.structure_description = types.MethodType(structure_description, None, PermutationGroup_generic)
+    from sage.groups.generic import structure_description
+
 
 
 class PermutationGroup_subgroup(PermutationGroup_generic):
diff --git a/src/sage/groups/perm_gps/permgroup_element.pxd b/src/sage/groups/perm_gps/permgroup_element.pxd
index ea02ea77ffb..e6004fc4d34 100644
--- a/src/sage/groups/perm_gps/permgroup_element.pxd
+++ b/src/sage/groups/perm_gps/permgroup_element.pxd
@@ -9,6 +9,7 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
     cdef Element _gap_element
     cdef __tuple
     cdef PermutationGroupElement _new_c(self)
+    cpdef _mul_(self, other)
     cpdef _gap_list(self)
     cpdef domain(self)
     cdef public __custom_name
diff --git a/src/sage/groups/semimonomial_transformations/semimonomial_transformation.pxd b/src/sage/groups/semimonomial_transformations/semimonomial_transformation.pxd
index 8ade44089fd..bd4f62d497f 100644
--- a/src/sage/groups/semimonomial_transformations/semimonomial_transformation.pxd
+++ b/src/sage/groups/semimonomial_transformations/semimonomial_transformation.pxd
@@ -5,3 +5,4 @@ cdef class SemimonomialTransformation(MultiplicativeGroupElement):
     cdef object perm, alpha
 
     cdef _new_c(self)
+    cpdef _mul_(self, other)
diff --git a/src/sage/homology/chain_complex.py b/src/sage/homology/chain_complex.py
index 38aa19d927b..bc8fe40a23e 100644
--- a/src/sage/homology/chain_complex.py
+++ b/src/sage/homology/chain_complex.py
@@ -237,7 +237,7 @@ def ChainComplex(data=None, base_ring=None, grading_group=None,
     if grading_group is None:
         grading_group = ZZ
     if degree_of_differential != 1 and degree != 1:
-        raise(ValueError, 'specify only one of degree_of_differential or degree, not both')
+        raise ValueError('specify only one of degree_of_differential or degree, not both')
     if degree_of_differential != 1:
         degree = degree_of_differential
     try:
diff --git a/src/sage/interfaces/expect.py b/src/sage/interfaces/expect.py
index 8825327628d..8070e602d26 100644
--- a/src/sage/interfaces/expect.py
+++ b/src/sage/interfaces/expect.py
@@ -652,7 +652,7 @@ def _local_tmpfile(self):
         A string that provides a temporary filename and is unique for the
         given interface.
 
-        TEST:
+        TESTS:
 
         The filename is cached::
 
diff --git a/src/sage/interfaces/fricas.py b/src/sage/interfaces/fricas.py
index ca794cee56d..3b368721554 100644
--- a/src/sage/interfaces/fricas.py
+++ b/src/sage/interfaces/fricas.py
@@ -850,7 +850,7 @@ def __getitem__(self, n):
 
             - can we somehow implement negative arguments?
 
-        TEST:
+        TESTS:
 
             sage: fricas("[1,2,3]")[0]                                          # optional - fricas
             1
diff --git a/src/sage/interfaces/gap.py b/src/sage/interfaces/gap.py
index fb05adc78b8..dc1bfcbeaa9 100644
--- a/src/sage/interfaces/gap.py
+++ b/src/sage/interfaces/gap.py
@@ -188,6 +188,7 @@
 from sage.structure.element import ModuleElement
 import re
 import os
+import io
 import pexpect
 import time
 import platform
@@ -1329,8 +1330,9 @@ def help(self, s, pager=True):
         else:
             tmp_to_use = self._local_tmpfile()
         self.eval('SetGAPDocTextTheme("none")')
-        self.eval(r'\$SAGE.tempfile := "%s";'%tmp_to_use)
-        line = Expect.eval(self, "? %s"%s)
+        gap_encoding = str(self('GAPInfo.TermEncoding;'))
+        self.eval(r'\$SAGE.tempfile := "%s";' % tmp_to_use)
+        line = Expect.eval(self, "? %s" % s)
         Expect.eval(self, "? 1")
         match = re.search("Page from (\d+)", line)
         if match is None:
@@ -1339,7 +1341,7 @@ def help(self, s, pager=True):
             (sline,) = match.groups()
             if self.is_remote():
                 self._get_tmpfile()
-            F = open(self._local_tmpfile(),"r")
+            F = io.open(self._local_tmpfile(), "r", encoding=gap_encoding)
             help = F.read()
             if pager:
                 from IPython.core.page import page
diff --git a/src/sage/interfaces/genus2reduction.py b/src/sage/interfaces/genus2reduction.py
index 9d529282584..1d56f694952 100644
--- a/src/sage/interfaces/genus2reduction.py
+++ b/src/sage/interfaces/genus2reduction.py
@@ -101,7 +101,7 @@ class is R, then the following is the meaning of
        the reduction (Namikawa, Ueno:"The complete classification of
        fibers in pencils of curves of genus two", Manuscripta Math., vol.
        9, (1973), pages 143-186.) The reduction symbol is followed by the
-       corresponding page number (or just an indiction) in the above
+       corresponding page number (or just an indication) in the above
        article. The lower index is printed by , for instance, [I2-II-5]
        means [I_2-II-5]. Note that if `K` and `K'` are
        Kodaira symbols for singular fibers of elliptic curves, [K-K'-m]
diff --git a/src/sage/interfaces/kash.py b/src/sage/interfaces/kash.py
index a0d8ad5c2c1..f8b16f4b3e8 100644
--- a/src/sage/interfaces/kash.py
+++ b/src/sage/interfaces/kash.py
@@ -520,7 +520,8 @@ def _start(self):
         try:
             Expect._start(self)
         except RuntimeError:
-            raise RuntimeError("You must install the optional Kash package to use Kash from Sage.")
+            from sage.misc.package import PackageNotFoundError
+            raise PackageNotFoundError("kash")
         # Turn off the annoying timer.
         self.eval('Time(false);')
 
diff --git a/src/sage/interfaces/lisp.py b/src/sage/interfaces/lisp.py
index 3abec1cf7ea..1913191fb1c 100644
--- a/src/sage/interfaces/lisp.py
+++ b/src/sage/interfaces/lisp.py
@@ -113,7 +113,7 @@ def eval(self, code, strip=True, **kwds):
             sage: lisp.eval('(+ 2 2)')
             '4'
 
-        TEST:
+        TESTS:
 
         Verify that it works when input == output::
 
@@ -158,7 +158,7 @@ def set(self, var, value):
             sage: lisp.get('x')
             '2'
 
-        TEST:
+        TESTS:
 
         It must also be possible to eval the variable by name::
 
diff --git a/src/sage/interfaces/macaulay2.py b/src/sage/interfaces/macaulay2.py
index 5e2cacec18d..0c979b88306 100644
--- a/src/sage/interfaces/macaulay2.py
+++ b/src/sage/interfaces/macaulay2.py
@@ -587,8 +587,8 @@ def use(self, R):
 
     def new_from(self, type, value):
         """
-        Returns a new Macaulay2Element of type type constructed from
-        value.
+        Return a new ``Macaulay2Element`` of type ``type`` constructed from
+        ``value``.
 
         EXAMPLES::
 
diff --git a/src/sage/interfaces/maxima_lib.py b/src/sage/interfaces/maxima_lib.py
index cf5a37e9628..4cf1a526f3d 100644
--- a/src/sage/interfaces/maxima_lib.py
+++ b/src/sage/interfaces/maxima_lib.py
@@ -112,7 +112,7 @@
 ecl_eval("(remprop 'mfactorial 'grind)") # don't use ! for factorials (#11539)
 ecl_eval("(setf $errormsg nil)")
 
-# the following is a direct adaption of the definition of "retrieve"
+# the following is a direct adaptation of the definition of "retrieve"
 # in the Maxima file macsys.lisp. This routine is normally responsible
 # for displaying a question and returning the answer. We change it to
 # throw an error in which the text of the question is included. We do
diff --git a/src/sage/interfaces/polymake.py b/src/sage/interfaces/polymake.py
index 30e51394be1..e028f3a14f5 100644
--- a/src/sage/interfaces/polymake.py
+++ b/src/sage/interfaces/polymake.py
@@ -411,7 +411,7 @@ def _keyboard_interrupt(self):
         """
         Interrupt a computation with <Ctrl-c>
 
-        TEST:
+        TESTS:
 
         For reasons that are not clear to the author, the following test
         is very flaky. Therefore, this test is marked as "not tested".
diff --git a/src/sage/interfaces/process.pyx b/src/sage/interfaces/process.pyx
index 8d572648457..beb926c2064 100644
--- a/src/sage/interfaces/process.pyx
+++ b/src/sage/interfaces/process.pyx
@@ -11,6 +11,7 @@ Utilities for subprocess management.
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 import errno
 import signal
diff --git a/src/sage/interfaces/r.py b/src/sage/interfaces/r.py
index 9db603f8eb3..2406d3b7f50 100644
--- a/src/sage/interfaces/r.py
+++ b/src/sage/interfaces/r.py
@@ -668,7 +668,11 @@ def library(self, library_name):
             ...
             ImportError: ...
         """
-        ret = self.eval('require("%s")'%library_name)
+        ret = self.eval('require("%s")' % library_name)
+        try:
+            ret = ret.decode('utf-8')
+        except UnicodeDecodeError:
+            ret = ret.decode('latin-1')
         # try hard to parse the message string in a locale-independent way
         if ' library(' in ret:       # locale-independent key-word
             raise ImportError("%s"%ret)
diff --git a/src/sage/interfaces/singular.py b/src/sage/interfaces/singular.py
index aa2a670bae7..5c5ca5231f4 100644
--- a/src/sage/interfaces/singular.py
+++ b/src/sage/interfaces/singular.py
@@ -1611,7 +1611,7 @@ def sage_global_ring(self):
         from sage.all import PolynomialRing
         # Meanwhile Singulars quotient rings are also of 'ring' type, not 'qring' as it was in the past.
         # To find out if a singular ring is a quotient ring or not checking for ring type does not help
-        # and instead of that we we check if the quotient ring is zero or not:
+        # and instead of that we check if the quotient ring is zero or not:
         if (singular.eval('ideal(basering)==0')=='1'):
             return PolynomialRing(BR, names=singular.eval('varstr(basering)'), order=termorder_from_singular(singular))
         P = PolynomialRing(BR, names=singular.eval('varstr(basering)'), order=termorder_from_singular(singular))
diff --git a/src/sage/interfaces/tides.py b/src/sage/interfaces/tides.py
index c050cd41a09..22cb60af2b3 100644
--- a/src/sage/interfaces/tides.py
+++ b/src/sage/interfaces/tides.py
@@ -502,14 +502,14 @@ def genfiles_mintides(integrator, driver, f, ics, initial, final, delta,
                 aa = 'XX[{}]'.format(l0.index(str(a))+len(var))
             else:
                 consta=True
-                aa = RR(a).str(truncate=False)
+                aa = RR(a).str()
             if str(b) in lv:
                 bb = 'XX[{}]'.format(lv.index(str(b)))
             elif str(b) in l0:
                 bb = 'XX[{}]'.format(l0.index(str(b))+len(var))
             else:
                 constb = True
-                bb = RR(b).str(truncate=False)
+                bb = RR(b).str()
             if consta:
                 oper += '_c'
                 if not oper=='div':
@@ -626,12 +626,12 @@ def genfiles_mintides(integrator, driver, f, ics, initial, final, delta,
     """%(n-1)
     outfile.write(auxstring)
     for i in range(len(ics)):
-        outfile.write('\tv[{}] = {} ; \n'.format(i, RR(ics[i]).str(truncate=False)))
-    outfile.write('\ttini = {} ;\n'.format(RR(initial).str(truncate=False)))
-    outfile.write('\ttend = {} ;\n'.format(RR(final).str(truncate=False)))
-    outfile.write('\tdt   = {} ;\n'.format(RR(delta).str(truncate=False)))
-    outfile.write('\ttolrel = {} ;\n'.format(RR(tolrel).str(truncate=False)))
-    outfile.write('\ttolabs = {} ;\n'.format(RR(tolabs).str(truncate=False)))
+        outfile.write('\tv[{}] = {} ; \n'.format(i, RR(ics[i]).str()))
+    outfile.write('\ttini = {} ;\n'.format(RR(initial).str()))
+    outfile.write('\ttend = {} ;\n'.format(RR(final).str()))
+    outfile.write('\tdt   = {} ;\n'.format(RR(delta).str()))
+    outfile.write('\ttolrel = {} ;\n'.format(RR(tolrel).str()))
+    outfile.write('\ttolabs = {} ;\n'.format(RR(tolabs).str()))
     outfile.write('\textern char ofname[500];')
     outfile.write('\tstrcpy(ofname, "'+ output +'");\n')
     outfile.write('\tminc_tides(v,VARS,p,PARS,tini,tend,dt,tolrel,tolabs);\n')
@@ -792,7 +792,7 @@ def genfiles_mpfr(integrator, driver, f, ics, initial, final, delta,
                 aa = 'par[{}]'.format(lp.index(sa))
             else:
                 consta=True
-                aa = RR(a).str(truncate=False)
+                aa = RR(a).str()
             if sb in lv:
                 bb = 'var[{}]'.format(lv.index(sb))
             elif sb in l0:
@@ -801,7 +801,7 @@ def genfiles_mpfr(integrator, driver, f, ics, initial, final, delta,
                 bb = 'par[{}]'.format(lp.index(sb))
             else:
                 constb=True
-                bb = RR(b).str(truncate=False)
+                bb = RR(b).str()
             if consta:
                 oper += '_c'
                 if not oper=='div':
@@ -933,24 +933,24 @@ def genfiles_mpfr(integrator, driver, f, ics, initial, final, delta,
     outfile.write('\tfor(i=0; i<npar; i++) mpfr_init2(p[i], TIDES_PREC);\n')
 
     for i in range(npar):
-        outfile.write('\tmpfr_set_str(p[{}], "{}", 10, TIDES_RND);\n'.format(i,RR(parameter_values[i]).str(truncate=False)))
+        outfile.write('\tmpfr_set_str(p[{}], "{}", 10, TIDES_RND);\n'.format(i,RR(parameter_values[i]).str()))
     outfile.write('\tint nvar = {};\n\tmpfr_t v[nvar];\n'.format(VAR))
     outfile.write('\tfor(i=0; i<nvar; i++) mpfr_init2(v[i], TIDES_PREC);\n')
     for i in range(len(ics)):
-        outfile.write('\tmpfr_set_str(v[{}], "{}", 10, TIDES_RND);\n'.format(i,RR(ics[i]).str(truncate=False)))
+        outfile.write('\tmpfr_set_str(v[{}], "{}", 10, TIDES_RND);\n'.format(i,RR(ics[i]).str()))
     outfile.write('\tmpfr_t tolrel, tolabs;\n')
     outfile.write('\tmpfr_init2(tolrel, TIDES_PREC); \n')
     outfile.write('\tmpfr_init2(tolabs, TIDES_PREC); \n')
-    outfile.write('\tmpfr_set_str(tolrel, "{}", 10, TIDES_RND);\n'.format(RR(tolrel).str(truncate=False)))
-    outfile.write('\tmpfr_set_str(tolabs, "{}", 10, TIDES_RND);\n'.format(RR(tolabs).str(truncate=False)))
+    outfile.write('\tmpfr_set_str(tolrel, "{}", 10, TIDES_RND);\n'.format(RR(tolrel).str()))
+    outfile.write('\tmpfr_set_str(tolabs, "{}", 10, TIDES_RND);\n'.format(RR(tolabs).str()))
 
     outfile.write('\tmpfr_t tini, dt; \n')
     outfile.write('\tmpfr_init2(tini, TIDES_PREC); \n')
     outfile.write('\tmpfr_init2(dt, TIDES_PREC); \n')
 
 
-    outfile.write('\tmpfr_set_str(tini, "{}", 10, TIDES_RND);;\n'.format(RR(initial).str(truncate=False)))
-    outfile.write('\tmpfr_set_str(dt, "{}", 10, TIDES_RND);\n'.format(RR(delta).str(truncate=False)))
+    outfile.write('\tmpfr_set_str(tini, "{}", 10, TIDES_RND);;\n'.format(RR(initial).str()))
+    outfile.write('\tmpfr_set_str(dt, "{}", 10, TIDES_RND);\n'.format(RR(delta).str()))
     outfile.write('\tint nipt = {};\n'.format(floor((final-initial)/delta)))
     outfile.write('\tFILE* fd = fopen("' + output + '", "w");\n')
     outfile.write('\tmp_tides_delta(function_iteration, NULL, nvar, npar, nfun, v, p, tini, dt, nipt, tolrel, tolabs, NULL, fd);\n')
diff --git a/src/sage/knots/link.py b/src/sage/knots/link.py
index f892586d272..1e0ef7509fe 100644
--- a/src/sage/knots/link.py
+++ b/src/sage/knots/link.py
@@ -154,7 +154,7 @@ class Link(object):
         L = Link([[1, 4, 2, 3], [4, 1, 3, 2]])
         sphinx_plot(L.plot())
 
-    We can construct links from from the braid group::
+    We can construct links from the braid group::
 
         sage: B = BraidGroup(4)
         sage: L = Link(B([-1, -1, -1, -2, 1, -2, 3, -2]))
diff --git a/src/sage/lfunctions/dokchitser.py b/src/sage/lfunctions/dokchitser.py
index 48c20befec4..8d68826bf8f 100644
--- a/src/sage/lfunctions/dokchitser.py
+++ b/src/sage/lfunctions/dokchitser.py
@@ -365,6 +365,15 @@ def num_coeffs(self, T=1):
             sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
             sage: L.num_coeffs()
             4
+
+        Verify that ``num_coeffs`` works with non-real spectral
+        parameters, e.g. for the L-function of the level 10 Maass form
+        with eigenvalue 2.7341055592527126::
+
+            sage: ev = 2.7341055592527126
+            sage: L = Dokchitser(conductor=10, gammaV=[ev*i, -ev*i],weight=2,eps=1)
+            sage: L.num_coeffs()
+            26
         """
         return Integer(self._gp_call_inst('cflength', T))
 
diff --git a/src/sage/lfunctions/zero_sums.pyx b/src/sage/lfunctions/zero_sums.pyx
index 95bd74505c1..c8d99a2e01d 100644
--- a/src/sage/lfunctions/zero_sums.pyx
+++ b/src/sage/lfunctions/zero_sums.pyx
@@ -1228,7 +1228,7 @@ cdef class LFunctionZeroSum_EllipticCurve(LFunctionZeroSum_abstract):
             if n<expt:
                 y += self._sincsquared_summand_1(n, t, ap, p, logp, thetap,
                                                  sqrtp, logq, thetaq, sqrtq, z)
-        # Now iterate only only over those n that are 1 or 5 mod 6
+        # Now iterate only over those n that are 1 or 5 mod 6
         n = 11
         # First: those n that are <= sqrt(bound)
         bound1 = c_exp(t/2)
@@ -1567,7 +1567,7 @@ cdef class LFunctionZeroSum_EllipticCurve(LFunctionZeroSum_abstract):
 
           - If True, the computation is first run with small and then
             successively larger Delta values up to max_Delta. If at any
-            point the computed bound is 0 (or 1 when when root_number is -1
+            point the computed bound is 0 (or 1 when root_number is -1
             or True), the computation halts and that value is returned;
             otherwise the minimum of the computed bounds is returned.
           - If False, the computation is run a single time with
diff --git a/src/sage/libs/eclib/mwrank.pyx b/src/sage/libs/eclib/mwrank.pyx
index f16a70ff19d..feb3447d207 100644
--- a/src/sage/libs/eclib/mwrank.pyx
+++ b/src/sage/libs/eclib/mwrank.pyx
@@ -18,7 +18,7 @@ EXAMPLES::
     sage: t
     [[1:2:1]]
 """
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 import os
 import sys
diff --git a/src/sage/libs/gap/element.pxd b/src/sage/libs/gap/element.pxd
index 36a12b024d5..ac6457fb571 100644
--- a/src/sage/libs/gap/element.pxd
+++ b/src/sage/libs/gap/element.pxd
@@ -44,7 +44,9 @@ cdef class GapElement(RingElement):
     cdef _initialize(self, parent, libGAP_Obj obj)
     cpdef _type_number(self)
     cpdef is_bool(self)
-
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
+    cpdef _mod_(self, right)
 
 cdef class GapElement_Integer(GapElement):
     pass
diff --git a/src/sage/libs/linkages/padics/mpz.pxi b/src/sage/libs/linkages/padics/mpz.pxi
index 1c3694c3761..8b6241a2e8b 100644
--- a/src/sage/libs/linkages/padics/mpz.pxi
+++ b/src/sage/libs/linkages/padics/mpz.pxi
@@ -133,6 +133,10 @@ cdef inline bint creduce(mpz_t out, mpz_t a, long prec, PowComputer_ prime_pow)
 
     - returns True if the reduction is zero; False otherwise.
     """
+    # The following could fail if the value returned by
+    # prime_pow.pow_mpz_t_tmp(prec) is zero. We could add a sig_on()/sig_off()
+    # to keep sage from crashing. This comes at a performance penalty, however.
+    # A correct implementation of prime_pow should never return zero.
     mpz_mod(out, a, prime_pow.pow_mpz_t_tmp(prec))
     return mpz_sgn(out) == 0
 
@@ -306,7 +310,7 @@ cdef inline int cinvert(mpz_t out, mpz_t a, long prec, PowComputer_ prime_pow) e
     success = mpz_invert(out, a, prime_pow.pow_mpz_t_tmp(prec))
     if not success:
         raise ZeroDivisionError
-    
+
 cdef inline int cmul(mpz_t out, mpz_t a, mpz_t b, long prec, PowComputer_ prime_pow) except -1:
     """
     Multiplication.
@@ -380,7 +384,7 @@ cdef inline bint cisone(mpz_t out, PowComputer_ prime_pow) except -1:
     - returns True if `a = 1`, and False otherwise.
     """
     return mpz_cmp_ui(out, 1) == 0
-    
+
 cdef inline bint ciszero(mpz_t out, PowComputer_ prime_pow) except -1:
     """
     Returns whether this element is equal to 0.
@@ -395,7 +399,7 @@ cdef inline bint ciszero(mpz_t out, PowComputer_ prime_pow) except -1:
     - returns True if `a = 0`, and False otherwise.
     """
     return mpz_cmp_ui(out, 0) == 0
-    
+
 cdef inline int cpow(mpz_t out, mpz_t a, mpz_t n, long prec, PowComputer_ prime_pow) except -1:
     """
     Exponentiation.
diff --git a/src/sage/libs/linkages/padics/unram_shared.pxi b/src/sage/libs/linkages/padics/unram_shared.pxi
index ac07bd435a1..8db30e40331 100644
--- a/src/sage/libs/linkages/padics/unram_shared.pxi
+++ b/src/sage/libs/linkages/padics/unram_shared.pxi
@@ -1,4 +1,6 @@
+cimport cython
 
+@cython.binding(True)
 def frobenius_unram(self, arithmetic=True):
     """
     Returns the image of this element under the Frobenius automorphism
@@ -65,6 +67,8 @@ def frobenius_unram(self, arithmetic=True):
         ans += update
     return ans << ppow
 
+
+@cython.binding(True)
 def norm_unram(self, base = None):
     """
     Return the absolute or relative norm of this element.
@@ -144,6 +148,8 @@ def norm_unram(self, base = None):
             norm_of_uniformizer = (-1)**self.parent().degree() * self.parent().defining_polynomial()[0]
         return self.parent().ground_ring()(self.unit_part().matrix_mod_pn().det()) * norm_of_uniformizer**self.valuation()
 
+
+@cython.binding(True)
 def trace_unram(self, base = None):
     """
     Return the absolute or relative trace of this element.
diff --git a/src/sage/libs/ntl/ntl_GF2EContext.pyx b/src/sage/libs/ntl/ntl_GF2EContext.pyx
index 6247b50549c..3728a2fe69e 100644
--- a/src/sage/libs/ntl/ntl_GF2EContext.pyx
+++ b/src/sage/libs/ntl/ntl_GF2EContext.pyx
@@ -12,6 +12,7 @@
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 include 'misc.pxi'
 include 'decl.pxi'
diff --git a/src/sage/libs/ntl/ntl_ZZ_pContext.pyx b/src/sage/libs/ntl/ntl_ZZ_pContext.pyx
index 53bee03f500..bd158cdc471 100644
--- a/src/sage/libs/ntl/ntl_ZZ_pContext.pyx
+++ b/src/sage/libs/ntl/ntl_ZZ_pContext.pyx
@@ -12,6 +12,7 @@
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 include 'misc.pxi'
 include 'decl.pxi'
diff --git a/src/sage/libs/ntl/ntl_ZZ_pEX.pyx b/src/sage/libs/ntl/ntl_ZZ_pEX.pyx
index af48ddaf12c..5ba292c9163 100644
--- a/src/sage/libs/ntl/ntl_ZZ_pEX.pyx
+++ b/src/sage/libs/ntl/ntl_ZZ_pEX.pyx
@@ -138,7 +138,7 @@ cdef class ntl_ZZ_pEX(object):
 
     def __reduce__(self):
         """
-        TEST:
+        TESTS:
         sage: c=ntl.ZZ_pEContext(ntl.ZZ_pX([1,1,1], 7))
         sage: a = ntl.ZZ_pE([3,2], c)
         sage: b = ntl.ZZ_pE([1,2], c)
@@ -152,7 +152,7 @@ cdef class ntl_ZZ_pEX(object):
         """
         Returns a string representation of self.
 
-        TEST:
+        TESTS:
         sage: c=ntl.ZZ_pEContext(ntl.ZZ_pX([1,1,1], 7))
         sage: a = ntl.ZZ_pE([3,2], c)
         sage: b = ntl.ZZ_pE([1,2], c)
@@ -168,7 +168,7 @@ cdef class ntl_ZZ_pEX(object):
         """
         Return a copy of self.
 
-        TEST:
+        TESTS:
         sage: c=ntl.ZZ_pEContext(ntl.ZZ_pX([1,1,1], 7))
         sage: a = ntl.ZZ_pE([3,2], c)
         sage: b = ntl.ZZ_pE([1,2], c)
diff --git a/src/sage/libs/ntl/ntl_lzz_p.pyx b/src/sage/libs/ntl/ntl_lzz_p.pyx
index c0998a4343f..1c957a10fe3 100644
--- a/src/sage/libs/ntl/ntl_lzz_p.pyx
+++ b/src/sage/libs/ntl/ntl_lzz_p.pyx
@@ -407,7 +407,7 @@ def make_zz_p(val, context):
     """
     For unpickling.
 
-    TEST:
+    TESTS:
         sage: f = ntl.zz_p(1, 12)
         sage: loads(dumps(f)) == f
         True
diff --git a/src/sage/libs/ntl/ntl_mat_GF2E.pyx b/src/sage/libs/ntl/ntl_mat_GF2E.pyx
index a49bdc21765..45058e07660 100644
--- a/src/sage/libs/ntl/ntl_mat_GF2E.pyx
+++ b/src/sage/libs/ntl/ntl_mat_GF2E.pyx
@@ -609,7 +609,7 @@ cdef class ntl_mat_GF2E(object):
 
     def image(self):
         """
-        The rows of X are computed as basis of A's row space.  X is is
+        The rows of X are computed as basis of A's row space.  X is
         row echelon form.
 
         EXAMPLES::
diff --git a/src/sage/libs/singular/function.pyx b/src/sage/libs/singular/function.pyx
index 2337514b04e..5930bc04fd0 100644
--- a/src/sage/libs/singular/function.pyx
+++ b/src/sage/libs/singular/function.pyx
@@ -929,7 +929,7 @@ cdef class Converter(SageObject):
 
         - ``to_convert`` - a Singular ``leftv``
 
-        TEST:
+        TESTS:
 
         Check that negative integers come through unscathed::
 
diff --git a/src/sage/libs/singular/polynomial.pyx b/src/sage/libs/singular/polynomial.pyx
index e82076d02b0..6ab52a58929 100644
--- a/src/sage/libs/singular/polynomial.pyx
+++ b/src/sage/libs/singular/polynomial.pyx
@@ -15,6 +15,7 @@ AUTHOR:
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 from cysignals.signals cimport sig_on, sig_off
 
diff --git a/src/sage/libs/singular/singular.pyx b/src/sage/libs/singular/singular.pyx
index fa8cd7ad8e5..27123898cb3 100644
--- a/src/sage/libs/singular/singular.pyx
+++ b/src/sage/libs/singular/singular.pyx
@@ -15,7 +15,7 @@ AUTHOR:
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 ###############################################################################
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 include "sage/libs/ntl/decl.pxi"
 
diff --git a/src/sage/manifolds/coord_func_symb.py b/src/sage/manifolds/coord_func_symb.py
index f16052dcaaa..337218dd0f6 100644
--- a/src/sage/manifolds/coord_func_symb.py
+++ b/src/sage/manifolds/coord_func_symb.py
@@ -1727,6 +1727,22 @@ def one(self):
             elt = self._chart.manifold().base_field().one()
         return self.element_class(self, elt)
 
+    def characteristic(self):
+        """
+        Return the characteristic of the function ring.
+
+        This is the characteristic of the base ring.
+
+        EXAMPLES::
+
+            sage: M = Manifold(2, 'M', structure='topological')
+            sage: X.<x,y> = M.chart()
+            sage: FR = X.function_ring()
+            sage: FR.characteristic()
+            0
+        """
+        return self._chart.manifold().base_field().characteristic()
+    
     def from_base_ring(self, r):
         """
         Return the canonical embedding of ``r`` into ``self``.
diff --git a/src/sage/manifolds/differentiable/affine_connection.py b/src/sage/manifolds/differentiable/affine_connection.py
index 0f6a47a72af..e9b5bbf4580 100644
--- a/src/sage/manifolds/differentiable/affine_connection.py
+++ b/src/sage/manifolds/differentiable/affine_connection.py
@@ -182,7 +182,7 @@ class AffineConnection(SageObject):
         sage: Df[:]
         [2*x, -2*y, 0]
 
-    The action of an affine connection connection on a scalar field must
+    The action of an affine connection on a scalar field must
     coincide with the differential::
 
         sage: Df == f.differential()
diff --git a/src/sage/manifolds/differentiable/chart.py b/src/sage/manifolds/differentiable/chart.py
index bf7b8d18c5d..a7a887aa18c 100644
--- a/src/sage/manifolds/differentiable/chart.py
+++ b/src/sage/manifolds/differentiable/chart.py
@@ -548,6 +548,104 @@ def restrict(self, subset, restrictions=None):
                     dom._top_frames.remove(resu._frame)
         return self._dom_restrict[subset]
 
+    def symbolic_velocities(self, left='D', right=None):
+        r"""
+        Return a list of symbolic variables ready to be used by the
+        user as the derivatives of the coordinate functions with respect
+        to a curve parameter (i.e. the velocities along the curve).
+        It may actually serve to denote anything else than velocities,
+        with a name including the coordinate functions.
+        The choice of strings provided as 'left' and 'right' arguments
+        is not entirely free since it must comply with Python
+        prescriptions.
+
+        INPUT:
+
+        - ``left`` -- (default: ``D``) string to concatenate to the left
+          of each coordinate functions of the chart
+        - ``right`` -- (default: ``None``) string to concatenate to the
+          right of each coordinate functions of the chart
+
+        OUTPUT:
+
+        - a list of symbolic expressions with the desired names
+
+        EXAMPLES:
+
+        Symbolic derivatives of the Cartesian coordinates of the
+        3-dimensional Euclidean space::
+
+            sage: R3 = Manifold(3, 'R3', start_index=1)
+            sage: cart.<X,Y,Z> = R3.chart()
+            sage: D = cart.symbolic_velocities(); D
+            [DX, DY, DZ]
+            sage: D = cart.symbolic_velocities(left='d', right="/dt"); D
+            Traceback (most recent call last):
+            ...
+            ValueError: The name "dX/dt" is not a valid Python
+             identifier.
+            sage: D = cart.symbolic_velocities(left='d', right="_dt"); D
+            [dX_dt, dY_dt, dZ_dt]
+            sage: D = cart.symbolic_velocities(left='', right="'"); D
+            Traceback (most recent call last):
+            ...
+            ValueError: The name "X'" is not a valid Python
+             identifier.
+            sage: D = cart.symbolic_velocities(left='', right="_dot"); D
+            [X_dot, Y_dot, Z_dot]
+            sage: R.<t> = RealLine()
+            sage: canon_chart = R.default_chart()
+            sage: D = canon_chart.symbolic_velocities() ; D
+            [Dt]
+
+        """
+
+        from sage.symbolic.ring import var
+
+        # The case len(self[:]) = 1 is treated apart due to the
+        # following fact.
+        # In the case of several coordinates, the argument of 'var' (as
+        # implemented below after the case len(self[:]) = 1) is a list
+        # of strings of the form ['Dx1', 'Dx2', ...] and not a unique
+        # string of the form 'Dx1 Dx2 ...'.
+        # Although 'var' is supposed to accept both syntaxes, the first
+        # one causes an error when it contains only one argument, due to
+        # line 784 of sage/symbolic/ring.pyx :
+        # "return self.symbol(name, latex_name=formatted_latex_name, domain=domain)"
+        # In this line, the first argument 'name' of 'symbol' is a list
+        # and not a string if the argument of 'var' is a list of one
+        # string (of the type ['Dt']), which causes error in 'symbol'.
+        # This might be corrected.
+        if len(self[:]) == 1:
+            string_vel = left + format(self[:][0]) # will raise an error
+            # in case left is not a string
+            if right is not None:
+                string_vel += right # will raise an error in case right
+                # is not a string
+
+            # If the argument of 'var' contains only one word, for
+            # instance:
+            # sage: var('Dt')
+            # then 'var' does not return a tuple containing one symbolic
+            # expression, but the symbolic expression itself.
+            # This is taken into account below in order to return a list
+            # containing one symbolic expression.
+            return [var(string_vel)]
+
+        list_strings_velocities = [left + format(coord_func)
+                                   for coord_func in self[:]] # will
+        # raise an error in case left is not a string
+
+        if right is not None:
+            list_strings_velocities = [string_vel + right for string_vel
+                                       in list_strings_velocities] # will
+            # raise an error in case right is not a string
+
+        return list(var(list_strings_velocities))
+
+
+
+
 #*****************************************************************************
 
 class RealDiffChart(DiffChart, RealChart):
diff --git a/src/sage/manifolds/differentiable/curve.py b/src/sage/manifolds/differentiable/curve.py
index 4c44dbcc919..47bc070cc7e 100644
--- a/src/sage/manifolds/differentiable/curve.py
+++ b/src/sage/manifolds/differentiable/curve.py
@@ -63,7 +63,7 @@ class DifferentiableCurve(DiffMap):
       the values being lists or tuples of `n` symbolic expressions of `t`,
       where `n` is the dimension of `M`
     - ``name`` -- (default: ``None``) string; symbol given to the curve
-    - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote the
+    - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote
       the curve; if none is provided, ``name`` will be used
     - ``is_isomorphism`` -- (default: ``False``) determines whether the
       constructed object is a diffeomorphism; if set to ``True``,
@@ -763,10 +763,8 @@ def plot(self, chart=None, ambient_coords=None, mapping=None, prange=None,
         """
         from sage.rings.infinity import Infinity
         from sage.misc.functional import numerical_approx
-        from sage.plot.graphics import Graphics
-        from sage.plot.line import line
         from sage.manifolds.chart import RealChart
-        from sage.manifolds.utilities import set_axes_labels
+
         #
         # Get the @options from kwds
         #
@@ -857,9 +855,62 @@ def plot(self, chart=None, ambient_coords=None, mapping=None, prange=None,
                                [numerical_approx( x[j].substitute(parameters) )
                                 for j in ind_pc] )
                 t += dt
+
+        return self._graphics(plot_curve, ambient_coords,
+                              thickness=thickness,
+                              aspect_ratio=aspect_ratio, color= color,
+                              style=style, label_axes=label_axes)
+
+
+
+    def _graphics(self, plot_curve, ambient_coords, thickness=1,
+                  aspect_ratio='automatic', color='red', style='-',
+                  label_axes=True):
+        r"""
+        Plot a 2D or 3D curve in a Cartesian graph with axes labeled by
+        the ambient coordinates; it is invoked by the methods
+        :meth:`plot` of
+        :class:`~sage.manifolds.differentiable.curve.DifferentiableCurve`,
+        and its subclasses
+        (:class:`~sage.manifolds.differentiable.integrated_curve.IntegratedCurve`,
+        :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedAutoparallelCurve`,
+        and
+        :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedGeodesic`).
+
+        TESTS::
+
+            sage: M = Manifold(2, 'R^2')
+            sage: X.<x,y> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: c = M.curve([cos(t), sin(t)], (t, 0, 2*pi), name='c')
+            sage: graph = c._graphics([[1,2], [3,4]], [x,y])
+            sage: graph._objects[0].xdata == [1,3]
+            True
+            sage: graph._objects[0].ydata == [2,4]
+            True
+            sage: graph._objects[0]._options['thickness'] == 1
+            True
+            sage: graph._extra_kwds['aspect_ratio'] == 'automatic'
+            True
+            sage: graph._objects[0]._options['rgbcolor'] == 'red'
+            True
+            sage: graph._objects[0]._options['linestyle'] == '-'
+            True
+            sage: l = [r'$'+latex(x)+r'$', r'$'+latex(y)+r'$']
+            sage: graph._extra_kwds['axes_labels'] == l
+            True
+
+        """
+
+        from sage.plot.graphics import Graphics
+        from sage.plot.line import line
+        from sage.manifolds.utilities import set_axes_labels
+
+
         #
         # The plot
         #
+        n_pc = len(ambient_coords)
         resu = Graphics()
         resu += line(plot_curve, color=color, linestyle=style,
                      thickness=thickness)
diff --git a/src/sage/manifolds/differentiable/integrated_curve.py b/src/sage/manifolds/differentiable/integrated_curve.py
new file mode 100644
index 00000000000..21286917cbb
--- /dev/null
+++ b/src/sage/manifolds/differentiable/integrated_curve.py
@@ -0,0 +1,3424 @@
+# -*- coding: utf-8 -*-
+r"""
+Integrated Curves and Geodesics in Manifolds
+
+Given a differentiable manifold `M`, an *integrated curve* curve in `M`
+is a differentiable curve constructed as a solution to a system of
+second order differential equations.
+
+Integrated curves are implemented by :class:`IntegratedCurve`, which the
+classes :class:`IntegratedAutoparallelCurve` and
+:class:`IntegratedGeodesic` inherit.
+
+.. RUBRIC:: Examples: A geodesic in hyperbolic Poincaré half-plane
+
+First declare a chart over the Poincaré half-plane::
+
+    sage: M = Manifold(2, 'M')
+    sage: X.<x,y> = M.chart('x y:(0,+oo)')
+
+Then declare the hyperbolic Poincaré metric::
+
+    sage: g = M.metric('g')
+    sage: g[0,0], g[1,1] = 1/y^2, 1/y^2
+    sage: g.display()
+    g = y^(-2) dx*dx + y^(-2) dy*dy
+
+Pick an initial point and an initial tangent vector::
+
+    sage: p = M((0,1), name='p')
+    sage: v = M.tangent_space(p)((1,3/2), name='v')
+    sage: v.display()
+    v = d/dx + 3/2 d/dy
+
+Declare a geodesic with such initial conditions, denoting ``t`` the
+corresponding affine parameter::
+
+    sage: t = var('t')
+    sage: c = M.integrated_geodesic(g, (t, 0, 10), v, name='c')
+
+Numerically integrate the geodesic::
+
+    sage: sol = c.solve()
+
+Plot the geodesic after interpolating the solution ``sol``, since it is
+required to plot::
+
+    sage: interp = c.interpolate()
+    sage: graph = c.plot_integrated()
+    sage: p_plot = p.plot(size=30, label_offset=0.07, fontsize=20)
+    sage: v_plot = v.plot(label_offset=0.05, fontsize=20)
+    sage: graph + p_plot + v_plot
+    Graphics object consisting of 5 graphics primitives
+
+.. PLOT::
+
+    M = Manifold(2, 'M')
+    X = M.chart('x y'); x, y = X[:]
+    g = M.metric('g')
+    g[0,0], g[1,1] = 1/y**2, 1/y**2
+    p = M((0,1), name='p')
+    v = M.tangent_space(p)((1,3/2), name='v')
+    t = var('t')
+    c = M.integrated_geodesic(g, (t, 0, 10), v, name='c')
+    sol = c.solve()
+    interp = c.interpolate()
+    graph = c.plot_integrated()
+    p_plot = p.plot(size=30, label_offset=0.07, fontsize=20)
+    v_plot = v.plot(label_offset=0.05, fontsize=20)
+    sphinx_plot(graph + p_plot + v_plot)
+
+AUTHORS:
+
+- Karim Van Aelst (2017): initial version
+
+"""
+
+#***********************************************************************
+#       Copyright (C) 2017 Karim Van Aelst <karim.van-aelst@obspm.fr>
+#
+#  Distributed under the terms of the GNU General Public License (GPL)
+#  as published by the Free Software Foundation; either version 2 of
+#  the License, or (at your option) any later version.
+#                  http://www.gnu.org/licenses/
+#***********************************************************************
+
+from __future__ import print_function
+from sage.symbolic.expression import Expression
+from sage.rings.infinity import Infinity
+from sage.calculus.desolvers import desolve_system_rk4
+from sage.calculus.desolvers import desolve_odeint
+from sage.manifolds.chart import Chart
+from sage.manifolds.differentiable.curve import DifferentiableCurve
+from sage.manifolds.differentiable.tangent_vector import TangentVector
+from sage.calculus.interpolation import Spline
+from sage.misc.decorators import options
+from sage.misc.functional import numerical_approx
+
+class IntegratedCurve(DifferentiableCurve):
+    r"""
+    Given a chart with coordinates denoted `(x_{1}, \ldots, x_{n})`,
+    an instance of :class:`IntegratedCurve` is a curve
+    `t \mapsto (x_{1}(t), \ldots, x_{n}(t))` constructed as a
+    solution to a system of second order differential equations
+    satisfied by the coordinate curves `t \mapsto x_{i}(t)`.
+
+    INPUT:
+
+    - ``parent`` --
+      :class:`~sage.manifolds.differentiable.manifold_homset.IntegratedCurveSet`
+      the set of curves `\mathrm{Hom_{integrated}}(I, M)` to which the
+      curve belongs
+    - ``equations_rhs`` -- list of the right-hand sides of the equations
+      on the velocities only (the term *velocity* referring to the
+      derivatives `d x_{i} / dt` of the coordinate curves)
+    - ``velocities`` -- list of the symbolic expressions used in
+      ``equations_rhs`` to denote the velocities
+    - ``curve_parameter`` -- symbolic expression used in
+      ``equations_rhs`` to denote the parameter of the curve (denoted
+      `t` in the descriptions above)
+    - ``initial_tangent_vector`` --
+      :class:`~sage.manifolds.differentiable.tangent_vector.TangentVector`
+      initial tangent vector of the curve
+    - ``chart`` -- (default: ``None``) chart on the manifold in
+      which the equations are given; if ``None`` the default chart
+      of the manifold is assumed
+    - ``name`` -- (default: ``None``) string; symbol given to the curve
+    - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to
+      denote the curve; if none is provided, ``name`` will be used
+
+    EXAMPLES:
+
+    Motion of a charged particle in an axial magnetic field linearly
+    increasing in time and exponentially decreasing in space:
+
+    .. MATH::
+
+        \mathbf{B}(t,\mathbf{x}) = \frac{B_{0}t}{T} \exp \left(
+        -\frac{ x_{1}^{2} + x_{2}^{2} }{ L^{2} } \right) \mathbf{e_{3}}.
+
+    Equations of motion are:
+
+    .. MATH::
+
+        \begin{aligned}
+        \ddot{x}_{1}(t) &= \frac{qB(t,\mathbf{x}(t))}{m} \dot{x}_{2}(t), \\
+        \ddot{x}_{2}(t) &= -\frac{qB(t, \mathbf{x}(t))}{m} \dot{x}_{1}(t), \\
+        \ddot{x}_{3}(t) &= 0.
+        \end{aligned}
+
+    Start with declaring a chart on a 3-dimensional manifold and the
+    symbolic expressions denoting the velocities and the various
+    parameters::
+
+        sage: M = Manifold(3, 'M', start_index=1)
+        sage: X.<x1,x2,x3> = M.chart()
+        sage: var('t B_0 m q L T')
+        (t, B_0, m, q, L, T)
+        sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+        sage: D = X.symbolic_velocities(); D
+        [Dx1, Dx2, Dx3]
+        sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+
+    Set the initial conditions::
+
+        sage: p = M.point((0,0,0), name='p')
+        sage: Tp = M.tangent_space(p)
+        sage: v = Tp((1,0,1))
+
+    Declare an integrated curve and display information relative to it::
+
+        sage: c = M.integrated_curve(eqns, D, (t, 0, 5), v, name='c',
+        ....:                                              verbose=True)
+        The curve was correctly set.
+        Parameters appearing in the differential system defining the
+         curve are [B_0, L, T, m, q].
+        sage: c
+        Integrated curve c in the 3-dimensional differentiable
+         manifold M
+        sage: sys = c.system(verbose=True)
+        Curve c in the 3-dimensional differentiable manifold M
+         integrated over the Real interval (0, 5) as a solution to the
+         following system, written with respect to
+         Chart (M, (x1, x2, x3)):
+        <BLANKLINE>
+        Initial point: Point p on the 3-dimensional differentiable
+         manifold M with coordinates [0, 0, 0] with respect to
+         Chart (M, (x1, x2, x3))
+        Initial tangent vector: Tangent vector at Point p on
+         the 3-dimensional differentiable manifold M with
+         components [1, 0, 1] with respect to Chart (M, (x1, x2, x3))
+        <BLANKLINE>
+        d(x1)/dt = Dx1
+        d(x2)/dt = Dx2
+        d(x3)/dt = Dx3
+        d(Dx1)/dt = B_0*Dx2*q*t*e^(-(x1^2 + x2^2)/L^2)/(T*m)
+        d(Dx2)/dt = -B_0*Dx1*q*t*e^(-(x1^2 + x2^2)/L^2)/(T*m)
+        d(Dx3)/dt = 0
+        <BLANKLINE>
+
+    Generate a solution of the system and an interpolation of this
+    solution::
+
+        sage: sol = c.solve(step=0.2,
+        ....:         parameters_values={B_0:1, m:1, q:1, L:10, T:1},
+        ....:         solution_key='carac time 1', verbose=True)
+        Performing 4th order Runge-Kutta integration with Maxima by
+         default...
+        Numerical integration completed.
+        <BLANKLINE>
+        Checking all points are in the chart domain...
+        All points are in the chart domain.
+        <BLANKLINE>
+        The resulting list of points was associated with the key
+         'carac time 1' (if this key already referred to a former
+         numerical solution, such a solution was erased).
+        sage: interp = c.interpolate(solution_key='carac time 1',
+        ....:                interpolation_key='interp 1', verbose=True)
+        Performing cubic spline interpolation by default...
+        Interpolation completed and associated with the key 'interp 1'
+         (if this key already referred to a former interpolation,
+         such an interpolation was erased).
+
+    Such an interpolation is required to evaluate the curve and the
+    vector tangent to the curve for any value of the curve parameter::
+
+        sage: p = c(1.9, verbose=True)
+        Evaluating point coordinates from the interpolation associated
+         with the key 'interp 1' by default...
+        sage: p
+        Point on the 3-dimensional differentiable manifold M
+        sage: p.coordinates()     # abs tol 1e-12
+        (1.3776707219621374, -0.9000776970132945, 1.9)
+        sage: v2 = c.tangent_vector_eval_at(4.3, verbose=True)
+        Evaluating tangent vector components from the interpolation
+         associated with the key 'interp 1' by default...
+        sage: v2
+        Tangent vector at Point on the 3-dimensional differentiable
+         manifold M
+        sage: v2[:]     # abs tol 1e-12
+        [-0.9303968397216424, -0.3408080563014475, 1.0000000000000004]
+
+    Plotting a numerical solution (with or without its tangent vector
+    field) also requires the solution to be interpolated at least once::
+
+        sage: c_plot_2d_1 = c.plot_integrated(ambient_coords=[x1, x2],
+        ....:               interpolation_key='interp 1', thickness=2.5,
+        ....:               display_tangent=True, plot_points=200,
+        ....:               plot_points_tangent=10, scale=0.5,
+        ....:               color='blue', color_tangent='red',
+        ....:               verbose=True)
+        A tiny final offset equal to 0.000251256281407035 was introduced
+         for the last point in order to safely compute it from the
+         interpolation.
+        sage: c_plot_2d_1
+        Graphics object consisting of 11 graphics primitives
+
+    .. PLOT::
+
+        M = Manifold(3, 'M')
+        X = M.chart('x1 x2 x3'); x1, x2, x3 = X[:]
+        t, B_0, m, q, L, T = var('t B_0 m q L T')
+        B = B_0*t/T*exp(-(x1**2 + x2**2)/L**2)
+        D = X.symbolic_velocities()
+        eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+        p = M.point((0,0,0), name='p')
+        Tp = M.tangent_space(p)
+        v = Tp((1,0,1))
+        c = M.integrated_curve(eqns, D, (t, 0, 5), v, name='c')
+        sol = c.solve(step=0.2,
+                      parameters_values={B_0:1, m:1, q:1, L:10, T:1},
+                      solution_key='carac time 1')
+        interp = c.interpolate(solution_key='carac time 1',
+                               interpolation_key='interp 1')
+        c_plot_2d_1 = c.plot_integrated(ambient_coords=[x1, x2],
+                        interpolation_key='interp 1', thickness=2.5,
+                        display_tangent=True, plot_points=200,
+                        plot_points_tangent=10, scale=0.5, color='blue',
+                        color_tangent='red')
+        sphinx_plot(c_plot_2d_1)
+
+    An instance of :class:`IntegratedCurve` may store several numerical
+    solutions and interpolations::
+
+        sage: sol = c.solve(step=0.2,
+        ....:         parameters_values={B_0:1, m:1, q:1, L:10, T:100},
+        ....:         solution_key='carac time 100')
+        sage: interp = c.interpolate(solution_key='carac time 100',
+        ....:                            interpolation_key='interp 100')
+        sage: c_plot_3d_100 = c.plot_integrated(interpolation_key='interp 100',
+        ....:                   thickness=2.5, display_tangent=True,
+        ....:                   plot_points=200, plot_points_tangent=10,
+        ....:                   scale=0.5, color='green',
+        ....:                   color_tangent='orange')
+        sage: c_plot_3d_1 = c.plot_integrated(interpolation_key='interp 1',
+        ....:                   thickness=2.5, display_tangent=True,
+        ....:                   plot_points=200, plot_points_tangent=10,
+        ....:                   scale=0.5, color='blue',
+        ....:                   color_tangent='red')
+        sage: c_plot_3d_1 + c_plot_3d_100
+        Graphics3d Object
+
+    .. PLOT::
+
+        M = Manifold(3, 'M')
+        X = M.chart('x1 x2 x3'); x1, x2, x3 = X[:]
+        t, B_0, m, q, L, T = var('t B_0 m q L T')
+        B = B_0*t/T*exp(-(x1**2 + x2**2)/L**2)
+        D = X.symbolic_velocities()
+        eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+        p = M.point((0,0,0), name='p')
+        Tp = M.tangent_space(p)
+        v = Tp((1,0,1))
+        c = M.integrated_curve(eqns, D, (t, 0, 5), v, name='c')
+        sol = c.solve(step=0.2, parameters_values={B_0:1, m:1, q:1, L:10, T:1},
+                      solution_key='carac time 1')
+        interp = c.interpolate(solution_key='carac time 1',
+                               interpolation_key='interp 1')
+        sol = c.solve(step=0.2, parameters_values={B_0:1, m:1, q:1, L:10, T:100},
+                      solution_key='carac time 100')
+        interp = c.interpolate(solution_key='carac time 100',
+                               interpolation_key='interp 100')
+        c_plot_3d_1 = c.plot_integrated(interpolation_key='interp 1',
+                        thickness=2.5, display_tangent=True,
+                        plot_points=200, plot_points_tangent=10,
+                        scale=0.5, color='blue', color_tangent='red')
+        c_plot_3d_100 = c.plot_integrated(interpolation_key='interp 100',
+                            thickness=2.5, display_tangent=True,
+                            plot_points=200, plot_points_tangent=10,
+                            scale=0.5, color='green',
+                            color_tangent='orange')
+        graph = c_plot_3d_1 + c_plot_3d_100
+        sphinx_plot(graph)
+
+    """
+
+    def __init__(self, parent, equations_rhs, velocities,
+                 curve_parameter, initial_tangent_vector, chart=None,
+                 name=None, latex_name=None, verbose=False):
+        r"""
+        Construct a curve defined by a system of second order
+        differential equations in the coordinate functions.
+
+        TESTS::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q, L, T] = var('t B_0 m q L T')
+            sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns + [x1], D, (t, 0, 5), v,
+            ....:                                              name='c')
+            Traceback (most recent call last):
+            ...
+            ValueError: number of equations should equal codomain
+             dimension
+            sage: c = M.integrated_curve(eqns, D + [x1], (t, 0, 5), v,
+            ....:                                              name='c')
+            Traceback (most recent call last):
+            ...
+            ValueError: number of velocities should equal codomain
+             dimension
+            sage: c = M.integrated_curve(eqns, D,(t,-oo,5), v, name='c')
+            Traceback (most recent call last):
+            ...
+            ValueError: both boundaries of the interval defining the
+             domain of a Homset of integrated curves need to be finite
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), x1, name='c')
+            Traceback (most recent call last):
+            ...
+            TypeError: x1 should be a tangent vector
+
+            sage: c = M.integrated_curve(eqns, D, (x1,0,5), v, name='c')
+            Traceback (most recent call last):
+            ...
+            ValueError: x1 should not be used as the curve parameter
+             since it also denotes a coordinate or a velocity
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c'); c
+            Integrated curve c in the 3-dimensional differentiable
+             manifold M
+            sage: TestSuite(c).run()
+
+
+        """
+
+        from sage.symbolic.ring import SR
+
+        # start with parent class method to initialize the four last
+        # arguments:
+        DifferentiableCurve.__init__(self, parent, name=name,
+                                     latex_name=latex_name)
+
+        # check argument 'parent': 't_min' and 't_max' below are only
+        # allowed to be either expressions of finite real values:
+        domain = self.domain()
+        t_min = domain.lower_bound()
+        t_max = domain.upper_bound()
+        if t_min == -Infinity or t_max == +Infinity:
+            raise ValueError("both boundaries of the interval " +
+                             "need to be finite")
+
+        codomain = self.codomain()
+
+        # check argument 'equations_rhs':
+        dim = codomain.dim()
+        if len(equations_rhs) != dim:
+            raise ValueError("number of equations should equal " +
+                             "codomain dimension")
+
+        # check the chart:
+        if chart is not None:
+            if chart not in codomain.atlas():
+                raise ValueError("{} should be a chart ".format(chart) +
+                                 "on the {}".format(codomain))
+        else:
+            chart = codomain.default_chart()
+
+        # check argument 'velocities':
+        if len(velocities) != dim:
+            raise ValueError("number of velocities should equal " +
+                             "codomain dimension")
+        # in particular, check that no velocity coincides with a
+        # coordinate:
+        for vel in velocities:
+            if vel in chart[:]:
+                str_error = "{} should not be used as a ".format(vel)
+                str_error += "velocity since it also denotes "
+                str_error += "a coordinate"
+                raise ValueError(str_error)
+
+        # check argument 'curve_parameter':
+        if not isinstance(curve_parameter, Expression):
+            raise TypeError("{} should be ".format(curve_parameter) +
+                             "a symbolic expression")
+        # in particular, check that it does not coincide with a
+        # coordinate or a velocity:
+        coords_vels = list(chart[:]) + list(velocities)
+        if curve_parameter in coords_vels:
+            str_error = "{} should not be used ".format(curve_parameter)
+            str_error += "as the curve parameter since it also denotes "
+            str_error += "a coordinate or a velocity"
+            raise ValueError(str_error)
+        # the various algorithms called in 'solve' method are in charge
+        # of raising errors about possibly remaining problems regarding
+        # 'curve_parameter'
+
+        # check argument 'initial_tangent_vector':
+        if not isinstance(initial_tangent_vector, TangentVector):
+            raise TypeError("{} ".format(initial_tangent_vector) +
+                            "should be a tangent vector")
+        # in particular, check that its base point sits in the domain
+        # of the chart (if its coordinates are explicitly given):
+        initial_pt = initial_tangent_vector.parent().base_point()
+        # line above retrieves the initial point as the base point of
+        # the tangent space to which the initial tangent vector belongs
+        initial_pt_coords = initial_pt.coordinates(chart)
+        i0 = chart.manifold().start_index()
+        for i in range(dim):
+            if not isinstance(initial_pt_coords[i], Expression):
+                coord_value = initial_pt_coords[i]
+                coord_min = chart.coord_bounds(i+i0)[0][0]
+                coord_max = chart.coord_bounds(i+i0)[1][0]
+                if coord_value <= coord_min or coord_value >= coord_max:
+                    raise ValueError("initial point should be in the " +
+                                     "domain of the chart")
+
+        # prepare attribute '_parameters':
+        announced_variables = set(coords_vels + [curve_parameter])
+        parameters = set()
+        # extract all the variables appearing in the equations:
+        for eqn in equations_rhs:
+            if isinstance(eqn, Expression): # some right hand sides
+            # might merely be real numbers and not expressions, so that
+            # they do not contain any variable, and method 'variables'
+            # could not be called on them
+                parameters = parameters.union(eqn.variables())
+        # remove the Expressions that should not be treated as
+        # parameters (i.e. the coordinate functions, the velocities and
+        # the curve parameter):
+        parameters = parameters.difference(announced_variables)
+        # extract all the variables appearing in the boundaries:
+        if isinstance(t_min, Expression):
+            parameters = parameters.union(t_min.variables())
+        if isinstance(t_max, Expression):
+            parameters = parameters.union(t_max.variables())
+        # extract all the variables appearing in the initial point
+        # coordinates:
+        for coord in initial_pt_coords:
+            if isinstance(coord,Expression):
+                parameters = parameters.union(coord.variables())
+        # extract all the variables appearing in the initial tangent
+        # vector components:
+        initial_coord_basis = chart.frame().at(initial_pt)
+        initial_tgt_vec_comps=initial_tangent_vector[initial_coord_basis,:]
+        for comp in initial_tgt_vec_comps:
+            if isinstance(comp, Expression):
+                parameters = parameters.union(comp.variables())
+
+        # check at this stage that no parameter coincides with a
+        # coordinate, a velocity, or the curve parameter; this would
+        # mean that an Expression used to denote either a bound, a
+        # coordinate of the initial point or a component of the initial
+        # tangent vector coincides with a coordinate, a velocity or the
+        # curve parameter (which would make no sense):
+        if len(parameters) != 0:
+            for param in parameters:
+                if param in announced_variables:
+                    str_error = "{} should not be used ".format(param)
+                    str_error += "as a parameter since it also denotes "
+                    str_error += "a coordinate, a velocity or the "
+                    str_error += "curve parameter"
+                    raise ValueError(str_error)
+
+        # define all attributes
+        self._equations_rhs = list(equations_rhs) # converts to list
+        # since might not already be a list (which is later required)
+        self._velocities = list(velocities) # converts to list
+        # since might not already be a list (which is later required)
+        self._curve_parameter = curve_parameter
+        self._initial_tangent_vector = initial_tangent_vector
+        self._chart = chart
+        self._parameters = parameters
+        self._ode_solver = None # if needed, becomes an instance of
+        # 'ode_solver', which performs most of the numerical integrations
+        # offered by method 'solve'
+        self._solutions = {} # dictionary containing all numerically
+        # computed lists of points of the curve, the keys being chosen
+        # by the user when calling method 'solve'
+        self._interpolations = {} # dictionary containing lists of
+        # interpolation objects, each interpolation object implementing
+        # the interpolation of one of the numerical coordinate curves,
+        # and the keys being chosen by the user when calling
+        # method 'interpolate'
+
+        if verbose:
+            print("The curve was correctly set.")
+            if self._parameters:
+                print("Parameters appearing in the differential " +
+                      "system defining the curve are " +
+                      "{}.".format(sorted(self._parameters, key=str)))
+            else:
+                print("No parameter appears in the differential " +
+                      "system defining the curve.")
+
+    def _repr_(self):
+        r"""
+        Return a string representation of ``self``.
+
+        TESTS::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q, L, T] = var('t B_0 m q L T')
+            sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t, 0, 5), v) ; c
+            Integrated curve in the 3-dimensional differentiable
+             manifold M
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c'); c
+            Integrated curve c in the 3-dimensional differentiable
+             manifold M
+
+        """
+
+        description = "Integrated curve "
+        if self._name is not None:
+            description += self._name + " "
+        description += "in the {}".format(self._codomain)
+        return description
+
+    def __reduce__(self):
+        r"""
+        Reduction function for the pickle protocole.
+
+        TESTS::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q, L, T] = var('t B_0 m q L T')
+            sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c')
+            sage: c.__reduce__()
+            (<class 'sage.manifolds.differentiable.manifold_homset.IntegratedCurveSet_with_category.element_class'>,
+             (Set of Morphisms from Real interval (0, 5) to
+              3-dimensional differentiable manifold M in Category of
+              homsets of subobjects of sets and topological spaces which
+              actually are integrated curves,
+              [B_0*Dx2*q*t*e^(-(x1^2 + x2^2)/L^2)/(T*m),
+               -B_0*Dx1*q*t*e^(-(x1^2 + x2^2)/L^2)/(T*m),
+               0],
+              [Dx1, Dx2, Dx3],
+              t,
+              Tangent vector at Point p on the 3-dimensional
+               differentiable manifold M,
+              Chart (M, (x1, x2, x3)),
+              'c',
+              'c'))
+
+        Test of pickling::
+
+            sage: loads(dumps(c))
+            Integrated curve c in the 3-dimensional differentiable
+             manifold M
+
+        """
+
+        return (type(self), (self.parent(), self._equations_rhs,
+                self._velocities, self._curve_parameter,
+                self._initial_tangent_vector, self._chart,
+                self._name, self._latex_name))
+
+    def system(self, verbose=False):
+        r"""
+        Provide a detailed description of the system defining the curve
+        and return the system defining it: chart, equations and initial
+        conditions.
+
+        INPUT:
+
+        - ``verbose`` -- (default: ``False``) prints a detailed
+          description of the curve
+
+        OUTPUT:
+
+        - list containing
+
+          * the equations
+          * the initial conditions
+          * the chart
+
+        EXAMPLES:
+
+        System defining an integrated curve::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q, L, T] = var('t B_0 m q L T')
+            sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c')
+            sage: sys = c.system(verbose=True)
+            Curve c in the 3-dimensional differentiable manifold M
+             integrated over the Real interval (0, 5) as a solution to
+             the following system, written with respect to
+             Chart (M, (x1, x2, x3)):
+            <BLANKLINE>
+            Initial point: Point p on the 3-dimensional differentiable
+             manifold M with coordinates [0, 0, 0] with respect to
+             Chart (M, (x1, x2, x3))
+            Initial tangent vector: Tangent vector at Point p on the
+             3-dimensional differentiable manifold M with
+             components [1, 0, 1] with respect to Chart (M, (x1, x2, x3))
+            <BLANKLINE>
+            d(x1)/dt = Dx1
+            d(x2)/dt = Dx2
+            d(x3)/dt = Dx3
+            d(Dx1)/dt = B_0*Dx2*q*t*e^(-(x1^2 + x2^2)/L^2)/(T*m)
+            d(Dx2)/dt = -B_0*Dx1*q*t*e^(-(x1^2 + x2^2)/L^2)/(T*m)
+            d(Dx3)/dt = 0
+            <BLANKLINE>
+            sage: sys_mute = c.system()
+            sage: sys_mute == sys
+            True
+
+        """
+
+        v0 = self._initial_tangent_vector
+        chart = self._chart
+
+        if verbose:
+            initial_tgt_space = v0.parent()
+            initial_pt = initial_tgt_space.base_point() # retrieves
+            # the initial point as the base point of the tangent space
+            # to which initial tangent vector belongs
+            initial_pt_coords = list(initial_pt.coordinates(chart))
+            # previous line converts to list since would otherwise be a
+            # tuple ; will raise error if coordinates in chart are not
+            # known
+
+            initial_coord_basis = chart.frame().at(initial_pt)
+            initial_tgt_vec_comps = v0[initial_coord_basis,:] # will
+            # raise error if components in coordinate basis are not
+            # known
+
+            description = "Curve "
+            if self._name is not None:
+                description += self._name + " "
+            description += "in the {} ".format(self.codomain())
+            description += "integrated over the "
+            description += "{} ".format(self.domain())
+            description += "as a solution to the following system, "
+            description += "written with respect to "
+            description += "{}:\n\n".format(chart)
+
+            description += "Initial point: {} ".format(initial_pt)
+            description += "with coordinates "
+            description += "{} ".format(initial_pt_coords)
+            description += "with respect to {}\n".format(chart)
+
+            description += "Initial tangent vector: {} ".format(v0)
+            description += "with components "
+            description +="{}".format(initial_tgt_vec_comps)
+            description += " with respect to {}\n\n".format(chart)
+
+            for coord_func,velocity in zip(chart[:],self._velocities):
+                description += "d({})/d{} = {}\n".format(coord_func,
+                                                  self._curve_parameter,
+                                                  velocity)
+
+            for velocity,eqn in zip(self._velocities,self._equations_rhs):
+                description += "d({})/d{} = {}\n".format(velocity,
+                                                  self._curve_parameter,
+                                                  eqn)
+
+            print(description)
+
+        return [self._equations_rhs, v0, chart]
+
+    def solve_analytical(self, verbose=False):
+        r"""
+        Solve the differential system defining ``self`` analytically.
+
+        Solve analytically the differential system defining a curve
+        using Maxima via Sage solver ``desolve_system``.
+        In case of success, the analytical expressions are added to the
+        dictionary of expressions representing the curve.
+        Pay attention to the fact that ``desolve_system`` only considers
+        initial conditions given at an initial parameter value equal to
+        zero, although the parameter range may not contain zero.
+        Yet, assuming that it does, values of the coordinates functions
+        at such zero initial parameter value are denoted by the name of
+        the coordinate function followed by the string ``"_0"``.
+
+        OUTPUT:
+
+        - list of the analytical expressions of the coordinate functions
+          (when the differential system could be solved analytically),
+          or boolean ``False`` (in case the differential system could
+          not be solved analytically)
+
+        EXAMPLES:
+
+        Analytical expression of the trajectory of a charged particle in
+        a uniform, stationary magnetic field::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q] = var('t B_0 m q')
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B_0/m*D[1], -q*B_0/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c')
+            sage: sys = c.system(verbose=True)
+            Curve c in the 3-dimensional differentiable manifold M
+             integrated over the Real interval (0, 5) as a solution to
+             the following system, written with respect to
+             Chart (M, (x1, x2, x3)):
+            <BLANKLINE>
+            Initial point: Point p on the 3-dimensional differentiable
+             manifold M with coordinates [0, 0, 0] with respect to
+             Chart (M, (x1, x2, x3))
+            Initial tangent vector: Tangent vector at Point p on the
+             3-dimensional differentiable manifold M with components
+             [1, 0, 1] with respect to Chart (M, (x1, x2, x3))
+            <BLANKLINE>
+            d(x1)/dt = Dx1
+            d(x2)/dt = Dx2
+            d(x3)/dt = Dx3
+            d(Dx1)/dt = B_0*Dx2*q/m
+            d(Dx2)/dt = -B_0*Dx1*q/m
+            d(Dx3)/dt = 0
+            <BLANKLINE>
+            sage: sol = c.solve_analytical()
+            sage: c.expr()
+            ((B_0*q*x1_0 - Dx2_0*m*cos(B_0*q*t/m) +
+               Dx1_0*m*sin(B_0*q*t/m) + Dx2_0*m)/(B_0*q),
+             (B_0*q*x2_0 + Dx1_0*m*cos(B_0*q*t/m) +
+              Dx2_0*m*sin(B_0*q*t/m) - Dx1_0*m)/(B_0*q),
+             Dx3_0*t + x3_0)
+
+        """
+
+        from sage.calculus.var import function
+        from sage.calculus.functional import diff
+        from sage.calculus.desolvers import desolve_system
+        from sage.symbolic.assumptions import assume, forget
+        from sage.symbolic.ring import var
+
+        dim = self.codomain().dim()
+        i0 = self.codomain().start_index()
+        des = self._velocities + self._equations_rhs
+        par = self._curve_parameter
+
+        for param in self._parameters:
+            assume(param != 0)
+
+        y = []
+        for i in range(2*dim):
+            name = "y{}".format(i+i0)
+            y += [function(name)(par)]
+
+        for i in range(dim):
+            vel = self._velocities[i]
+            des[i] = des[i].substitute({vel: y[dim+i]})
+            des[i] = diff(y[i],par) == des[i]
+            for j in range(dim):
+                coord = self._chart[:][j] # important to use '[:]' on
+                # 'chart' to avoid problems due to non zero starting
+                # index (i0)
+                veloc = self._velocities[j]
+                des[dim+i] = des[dim+i].substitute({coord: y[j]})
+                des[dim+i] = des[dim+i].substitute({veloc: y[dim+j]})
+            des[dim+i] = (diff(y[dim+i], par) == des[dim+i])
+
+        dvars = y
+        ics = [0]
+        y_ics_first_half = []
+        y_ics_second_half = []
+        for i in range(dim):
+            coord = self._chart[:][i] # important to use '[:]'
+            # on 'chart' to avoid problems due to non zero
+            # starting index (i0)
+            veloc = self._velocities[i]
+            str_var_coord = "{}_0".format(coord)
+            str_var_veloc = "{}_0".format(veloc)
+            y_coord_0 = var(str_var_coord)
+            y_veloc_0 = var(str_var_veloc)
+            y_ics_first_half += [y_coord_0]
+            y_ics_second_half += [y_veloc_0]
+        ics += y_ics_first_half + y_ics_second_half
+
+        try:
+            sol = desolve_system(des, dvars, ivar=self._curve_parameter, ics=ics)
+        except NotImplementedError:
+            coords_sol_expr = False
+            if verbose:
+                print("The system could not be solved analytically.")
+        else:
+            coords_sol_expr = []
+            for relation in sol[:dim]:
+                expr = relation.rhs().simplify_full()
+                coords_sol_expr += [expr]
+            self.add_expr(self.domain().default_chart(), self._chart,
+                                                        coords_sol_expr)
+
+        for param in self._parameters:
+            forget(param != 0)
+
+        return tuple(coords_sol_expr)
+
+    def solve(self, step=None, method=None, solution_key=None,
+              parameters_values=None, verbose=False):
+        r"""
+        Integrate the curve numerically over the domain of integration.
+
+        INPUT:
+
+        - ``step`` -- (default: ``0.1``) step of integration
+        - ``method`` -- (default: ``None``) numerical scheme to use for
+          the integration of the curve; algorithms available are:
+
+          * ``'rk4_maxima'`` - 4th order classical Runge-Kutta, which
+            makes use of Maxima's dynamics package via Sage solver
+            ``desolve_system_rk4``
+          * ``'ode_int'`` - makes use of ``odeint`` from ``scipy.integrate``
+            module via Sage solver ``desolve_odeint``
+
+        and those provided by ``GSL`` via Sage class
+        :class:`~sage.calculus.ode.ode_solver`:
+
+          * ``'rk2'`` - embedded Runge-Kutta (2,3)
+          * ``'rk4'`` - 4th order classical Runge-Kutta
+          * ``'rkf45'`` - Runge-Kutta-Felhberg (4,5)
+          * ``'rkck'`` - embedded Runge-Kutta-Cash-Karp (4,5)
+          * ``'rk8pd'`` - Runge-Kutta prince-dormand (8,9)
+          * ``'rk2imp'`` - implicit 2nd order Runge-Kutta at Gaussian points
+          * ``'rk4imp'`` - implicit 4th order Runge-Kutta at Gaussian points
+          * ``'gear1'`` - `M=1` implicit Gear
+          * ``'gear2'`` - `M=2` implicit Gear
+          * ``'bsimp'`` - implicit Burlisch-Stoer (requires Jacobian)
+
+        - ``solution_key`` -- (default: ``None``) key which the
+          resulting numerical solution will be associated to; a default
+          value is given if none is provided
+        - ``parameters_values`` -- (default: ``None``) list of numerical
+          values of the parameters present in the system defining the
+          curve, to be substituted in the equations before integration
+        - ``verbose`` -- (default: ``False``) prints information about
+          the computation in progress
+
+        OUTPUT:
+
+        - list of the numerical points of the solution computed
+
+        EXAMPLES:
+
+        Computing a numerical solution::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q, L, T] = var('t B_0 m q L T')
+            sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c')
+            sage: sol = c.solve(parameters_values={m:1, q:1, L:10, T:1})
+            Traceback (most recent call last):
+            ...
+            ValueError: numerical values should be provided for each of
+             the parameters [B_0, L, T, m, q]
+            sage: sol = c.solve(method='my method',
+            ....:        parameters_values={B_0:1, m:1, q:1, L:10, T:1})
+            Traceback (most recent call last):
+            ...
+            ValueError: no available method of integration referred to
+             as 'my method'
+            sage: sol = c.solve(
+            ....:        parameters_values={B_0:1, m:1, q:1, L:10, T:1},
+            ....:        verbose=True)
+            Performing 4th order Runge-Kutta integration with Maxima by
+             default...
+            Resulting list of points will be associated with the key
+             'rk4_maxima' by default.
+            Numerical integration completed.
+            <BLANKLINE>
+            Checking all points are in the chart domain...
+            All points are in the chart domain.
+            <BLANKLINE>
+            The resulting list of points was associated with the key
+             'rk4_maxima' (if this key already referred to a former
+             numerical solution, such a solution was erased).
+            sage: sol_mute = c.solve(
+            ....:        parameters_values={B_0:1, m:1, q:1, L:10, T:1})
+            sage: sol_mute == sol
+            True
+
+        """
+
+        from sage.symbolic.ring import SR
+
+        if method is None:
+            method = 'rk4_maxima'
+            if verbose:
+                print("Performing 4th order Runge-Kutta integration " +
+                      "with Maxima by default...")
+
+        if solution_key is None:
+            solution_key = method
+            if verbose:
+                print("Resulting list of points will be associated " +
+                      "with the key '{}' ".format(solution_key) +
+                      "by default.")
+
+        t_min = self.domain().lower_bound()
+        t_max = self.domain().upper_bound()
+
+        eqns_num = [eq for eq in self._equations_rhs]
+        # 'self._equations_rhs' needs not to be modified ever, because we
+        # want to keep track of the most general form of the equations
+        # defining self, since those may contain parameters (which, for
+        # instance, we want to display as their original expressions
+        # when calling 'system' method with option 'verbose', and not
+        # substituted with some numerical values).
+        # This is why 'eqns_num' is declared: it will contain copies of
+        # the equations of 'self._equations_rhs' in which the parameters
+        # will be substituted with numerical values.
+        # It was then important to declare it as above, in order to make
+        # independent copies of each equations of 'self._equations_rhs',
+        # rather than declaring 'eqns_num = self._equations_rhs', in which
+        # case making substitutions in 'eqns_num' would have meant making
+        # the same substitutions in the original equations of
+        # 'self._equations_rhs'
+
+        v0 = self._initial_tangent_vector
+        chart = self._chart
+
+        initial_tgt_space = v0.parent()
+        initial_pt = initial_tgt_space.base_point() # retrieves
+        # the initial point as the base point of the tangent space
+        # to which the initial tangent vector belongs
+        initial_pt_coords = list(initial_pt.coordinates(chart))
+        # previous line converts to list since would otherwise be a
+        # tuple (yet might need to be added to [t_min] later); will
+        # raise error if coordinates in chart cannot be obtained
+
+        initial_coord_basis = chart.frame().at(initial_pt)
+        initial_tgt_vec_comps = list(v0[initial_coord_basis,:]) #idem
+
+        dim = self.codomain().dim()
+
+        if self._parameters:
+            if parameters_values is None or len(parameters_values) != len(self._parameters):
+                raise ValueError("numerical values should be " +
+                                 "provided for each of the " +
+                                 "parameters "
+                                 "{}".format(sorted(self._parameters, key=str)))
+            for key in parameters_values:
+                # Get numerical values in case some parameters values
+                # contain expressions such as pi; will raise error if
+                # any element of parameters_values is not numerical
+                parameters_values[key] = numerical_approx(parameters_values[key])
+
+            if isinstance(t_min, Expression):
+                t_min = parameters_values[t_min]
+                if t_min == -Infinity or t_min == +Infinity:
+                    raise ValueError("both boundaries of the " +
+                                      "interval need to be finite")
+
+            if isinstance(t_max, Expression):
+                t_max = parameters_values[t_max]
+                if t_max == -Infinity or t_max == +Infinity:
+                    raise ValueError("both boundaries of the " +
+                                     "interval need to be finite")
+
+            for i in range(dim):
+                if isinstance(eqns_num[i], Expression): # some right
+                # hand sides might merely be real numbers and not
+                # expressions, so that they do not contain any variable,
+                # and hence no substitution is required
+                    eqns_num[i] = eqns_num[i].substitute(parameters_values)
+
+            for i in range(dim):
+                if isinstance(initial_pt_coords[i], Expression):
+                    AUX = initial_pt_coords[i]
+                    AUX = AUX.substitute(parameters_values)
+                    initial_pt_coords[i] = AUX
+                if isinstance(initial_tgt_vec_comps[i], Expression):
+                    AUX2 = initial_tgt_vec_comps[i]
+                    AUX2 = AUX2.substitute(parameters_values)
+                    initial_tgt_vec_comps[i] = AUX2
+                # 'AUX' and 'AUX2' only used for the lines of
+                # source code to be shorter
+
+        t_min = numerical_approx(t_min)
+        t_max = numerical_approx(t_max)
+
+        for i in range(dim):
+            if not isinstance(eqns_num[i], Expression): # in case of a
+            # right hand side that is not an Expression (and then is a
+            # number), it is needed to be converted to an Expression
+            # since some solvers called below require only expressions
+                eqns_num[i] = SR(eqns_num[i])
+
+        if step is None:
+            step = (t_max - t_min) / 100
+
+        initial_pt_coords = [numerical_approx(coord) for coord
+                             in initial_pt_coords]
+        initial_tgt_vec_comps = [numerical_approx(comp) for comp
+                                 in initial_tgt_vec_comps]
+        # the last two instructions retrieve numerical values even
+        # if no parameters had to be substituted, in case some
+        # coordinates or components contain expressions such as pi,
+        # or are not RealNumber, since variable 'ics' of
+        # 'desolve_system_rk4' used below needs to be a list of
+        # RealNumber
+
+        if not chart.valid_coordinates(*initial_pt_coords):
+            raise ValueError("initial point should be in the " +
+                             "domain of the chart")
+
+        ode_solver_methods = ["rk2", "rk4", "rkf45", "rkck", "rk8pd",
+                              "rk2imp", "rk4imp", "gear1", "gear2", "bsimp"]
+
+        if method == 'rk4_maxima':
+            des = self._velocities + eqns_num
+            dvars = list(chart[:]) + self._velocities
+            ics = [t_min] + initial_pt_coords + initial_tgt_vec_comps
+
+            sol = desolve_system_rk4(des, dvars,
+                                     ivar=self._curve_parameter,
+                                     ics=ics,
+                                     end_points=[t_min, t_max],
+                                     step=step)
+
+            # The value of 'step' being set by the user when calling
+            # method 'solve', the value of (t_max - tmin)/step is not
+            # necessarily an integer.
+            # As a result, when the solver 'desolve_system_rk4' reaches
+            # a curve parameter that is distant to 't_max' by less than
+            # 'step', it computes one last point evaluated for a curve
+            # parameter exactly equal to 't_max'.
+            # Therefore, the difference between the curve parameter
+            # corresponding to this last point and that corresponding
+            # to the previous one is strictly less than 'step'. If this
+            # difference is too small (that is, if the solver considered
+            # that it did not reach 't_max', and hence computed one more
+            # point, although it was already very close to 't_max'),
+            # problems arise when using an interpolation of this
+            # solution (such as getting points with coordinates 'nan').
+            # As a result, we choose to remove the last point of a
+            # solution when it is a point that was added by the solver
+            # and threatens to be too close to the previous one
+            # (arbitrarily, we consider two points to be too close if
+            # their curve parameters are separated by less than 90% of a
+            # step).
+            if len(sol) > 1 and abs(sol[-1][0] - sol[-2][0]) < 0.9 * step:
+                del sol[-1]
+
+        elif method == "ode_int":
+            des = self._velocities + eqns_num
+            ics = initial_pt_coords + initial_tgt_vec_comps
+            times = srange(t_min, t_max, step, include_endpoint=True)
+            dvars = list(chart[:]) + self._velocities
+
+            sol0 = desolve_odeint(des, ics, times, dvars,
+                                             ivar=self._curve_parameter)
+
+            # rewrite the solution to prepare for the extraction (which
+            # removes information about the velocities), and convert
+            # elements of type 'numpy.float64' to standard type 'float'
+            sol = []
+            for t, coords_array in zip(times, sol0):
+                coords_values = [float(coord_value) for coord_value
+                                 in coords_array]
+                sol += [ [t] + coords_values ]
+        elif method in ode_solver_methods:
+            T = self._ode_solver
+
+            if T is None:
+                def system(t, y):
+                    syst = self._velocities + eqns_num
+                    par = self._curve_parameter
+                    for i in range(dim):
+                        vel = self._velocities[i]
+                        syst[i] = syst[i].substitute({vel:y[dim+i]})
+                        syst[dim+i] = syst[dim+i].substitute({par:t})
+                        for j in range(dim):
+                            coord = chart[:][j] # important to use '[:]'
+                            # on 'chart' to avoid problems due to non
+                            # zero starting index (i0)
+                            veloc = self._velocities[j]
+                            syst[dim+i] = syst[dim+i].substitute({coord:y[j]})
+                            syst[dim+i] = syst[dim+i].substitute({veloc:y[dim+j]})
+                    return syst
+                from sage.calculus.ode import ode_solver
+                T = ode_solver(function=system)
+
+            T.algorithm = method
+            y_0 = initial_pt_coords + initial_tgt_vec_comps
+            t_span = srange(t_min, t_max, step, include_endpoint=True)
+
+            if method == "bsimp":
+                # this method requires the expression of the Jacobian
+                # matrix of the application defining the right-hand side
+                # of the system to be provided
+
+                if T.jacobian is None:
+                    def jacobian(t,y):
+                        jac = []
+                        par = self._curve_parameter
+                        for i in range(dim):
+                            new_row = [0] * (2*dim)
+                            new_row[dim + i] = 1
+                            jac += [new_row]
+
+                        for i in range(dim):
+                            semi_row_coords = []
+                            semi_row_vels = []
+                            for j in range(dim):
+                                coord = chart[:][j] # important to use
+                                # '[:]' on 'chart' to avoid problems due
+                                # to non zero starting index (i0)
+                                vel = self._velocities[j]
+                                AUX = eqns_num[i].derivative(coord)
+                                AUX2 = eqns_num[i].derivative(vel)
+                                AUX = AUX.substitute({par: t})
+                                AUX2 = AUX2.substitute({par: t})
+                                for k in range(dim):
+                                    coordin = chart[:][k] # important to
+                                    # use '[:]' on 'chart' to avoid
+                                    # problems due to non zero starting
+                                    # index (i0)
+                                    veloc = self._velocities[k]
+                                    AUX = AUX.substitute({coordin: y[k]})
+                                    AUX = AUX.substitute({veloc: y[dim+k]})
+                                    AUX2 = AUX2.substitute({coordin: y[k]})
+                                    AUX2 = AUX2.substitute({veloc: y[dim+k]})
+                                semi_row_coords += [AUX]
+                                semi_row_vels += [AUX2]
+                            jac += [semi_row_coords + semi_row_vels]
+
+                        last_semi_row_coords = [0] * dim
+                        last_semi_row_vels = []
+                        for j in range(dim):
+                            AUX3 = eqns_num[j].derivative(par)
+                            AUX3 = AUX3.substitute({par: t})
+                            for m in range(dim):
+                                coordin = chart[:][m] # important to use
+                                # '[:]' on 'chart' to avoid problems due
+                                # to non zero starting index (i0)
+                                veloc = self._velocities[m]
+                                AUX3 = AUX3.substitute({coordin: y[m]})
+                                AUX3 = AUX3.substitute({veloc: y[dim+m]})
+                            last_semi_row_vels += [AUX3]
+                        jac += [last_semi_row_coords + last_semi_row_vels]
+                        # 'AUX', 'AUX2' and 'AUX3' only used for the lines
+                        # of source code to be shorter
+                        return jac
+                    T.jacobian = jacobian
+
+                T.ode_solve(jacobian=jacobian, y_0=y_0, t_span=t_span)
+            else:
+                T.ode_solve(y_0=y_0, t_span=t_span)
+
+            sol0 = T.solution
+            sol = []
+            for point in sol0:
+                sol += [[point[0]] + point[1]]
+            # above loop rewrites the solution in the same form than
+            # that provided by other methods ('rk4_maxima' and
+            # 'ode_int'), in order to extract the time and corresponding
+            # coordinate values a few lines below, in the same way for
+            # all methods
+
+        else:
+            raise ValueError("no available method of integration " +
+                             "referred to as '{}'".format(method))
+
+        # eventually, extract the time and corresponding coordinate
+        # values from each point of the solution computed (thus removing
+        # information about the values of the velocities ; should the
+        # latter be conserved ? They could turn useful in method
+        # 'tangent_vector_eval_at', and in 'plot' when plotting the
+        # tangent vectors.)
+        coords_sol = [point[0:dim+1] for point in sol]
+
+        if verbose:
+            print("Numerical integration completed.\n\n" +
+                  "Checking all points are in the chart domain...")
+
+        N = len(coords_sol)
+        n = 0
+        while n < N and chart.valid_coordinates(*coords_sol[n][1:dim+1]):
+            n += 1
+
+        if n < N:
+            raise ValueError("the {}th point ".format(n) +
+                             "(initial point being the '0th' point) " +
+                             "of the numerical solution (obtained " +
+                             "for a curve parameter equal " +
+                             "to {}) is out ".format(sol[n][0]) +
+                             "of the chart domain; a curve with a " +
+                             "smaller maximal value of the curve " +
+                             "parameter, or a smaller initial tangent "+
+                             "vector might be considered")
+        else:
+            self._solutions[solution_key] = coords_sol
+            if verbose:
+                print("All points are in the chart domain.\n\n" +
+                      "The resulting list of points was associated " +
+                      "with the key '{}' ".format(solution_key) +
+                      "(if this key already referred to a former " +
+                      "numerical solution, such a solution was erased).")
+            return self._solutions[solution_key]
+
+    def solution(self, solution_key=None, verbose=False):
+        r"""
+        Return the solution (list of points) associated with the given
+        key.
+
+        INPUT:
+
+        - ``solution_key`` -- (default: ``None``) key which the
+          requested numerical solution is associated to; a default
+          value is chosen if none is provided
+        - ``verbose`` -- (default: ``False``) prints information about
+          the solution returned
+
+        OUTPUT:
+
+        - list of the numerical points of the solution requested
+
+        EXAMPLES:
+
+        Requesting a numerical solution previously computed::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q, L, T] = var('t B_0 m q L T')
+            sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c')
+            sage: sol = c.solve(solution_key='sol_T1',
+            ....:        parameters_values={B_0:1, m:1, q:1, L:10, T:1})
+            sage: sol_bis = c.solution(verbose=True)
+            Returning the numerical solution associated with the key
+             'sol_T1' by default...
+            sage: sol_bis == sol
+            True
+            sage: sol_ter = c.solution(solution_key='sol_T1')
+            sage: sol_ter == sol
+            True
+            sage: sol_mute = c.solution()
+            sage: sol_mute == sol
+            True
+
+        """
+
+        if solution_key is None:
+            if 'rk4_maxima' in self._solutions:
+                solution_key = 'rk4_maxima'
+            else:
+                solution_key = self._solutions.keys()[0] # will raise
+                # error if self._solutions empty
+            if verbose:
+                print("Returning the numerical solution associated " +
+                      "with the key '{}' ".format(solution_key) +
+                      "by default...")
+        elif solution_key not in self._solutions:
+            raise ValueError("no existing key " +
+                             "'{}' ".format(solution_key) +
+                             "referring to any numerical solution")
+
+        return self._solutions[solution_key]
+
+    def interpolate(self, solution_key=None, method=None,
+                    interpolation_key=None, verbose=False):
+        r"""
+        Interpolate the chosen numerical solution using the given
+        interpolation method.
+
+        INPUT:
+
+        - ``solution_key`` -- (default: ``None``) key which the
+          numerical solution to interpolate is associated to ; a default
+          value is chosen if none is provided
+        - ``method`` -- (default: ``None``) interpolation scheme to use;
+          algorithms available are
+
+          * ``'cubic spline'``, which makes use of ``GSL`` via
+            :class:`~sage.calculus.interpolation.Spline`
+
+        - ``interpolation_key`` -- (default: ``None``) key which the
+          resulting interpolation will be associated to ; a default
+          value is given if none is provided
+        - ``verbose`` -- (default: ``False``) prints information about
+          the interpolation in progress
+
+        OUTPUT:
+
+        - built interpolation object
+
+        EXAMPLES:
+
+        Interpolating a numerical solution previously computed::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q, L, T] = var('t B_0 m q L T')
+            sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c')
+            sage: sol = c.solve(method='rk4_maxima',
+            ....:        solution_key='sol_T1',
+            ....:        parameters_values={B_0:1, m:1, q:1, L:10, T:1})
+            sage: interp = c.interpolate(solution_key='my solution')
+            Traceback (most recent call last):
+            ...
+            ValueError: no existing key 'my solution' referring to any
+             numerical solution
+            sage: interp = c.interpolate(solution_key='sol_T1',
+            ....:                        method='my method')
+            Traceback (most recent call last):
+            ...
+            ValueError: no available method of interpolation referred to
+             as 'my method'
+            sage: interp = c.interpolate(method='cubic spline',
+            ....:                        solution_key='sol_T1',
+            ....:                        interpolation_key='interp_T1',
+            ....:                        verbose=True)
+            Interpolation completed and associated with the key
+             'interp_T1' (if this key already referred to a former
+             interpolation, such an interpolation was erased).
+            sage: interp = c.interpolate(verbose=True)
+            Interpolating the numerical solution associated with the
+             key 'sol_T1' by default...
+            Performing cubic spline interpolation by default...
+            Resulting interpolation will be associated with the key
+             'cubic spline-interp-sol_T1' by default.
+            Interpolation completed and associated with the key
+             'cubic spline-interp-sol_T1' (if this key already referred
+             to a former interpolation, such an interpolation was
+             erased).
+
+        """
+
+        if solution_key is None:
+            if 'rk4_maxima' in self._solutions:
+                solution_key = 'rk4_maxima'
+            else:
+                solution_key = self._solutions.keys()[0] # will raise
+                # error if self._solutions empty
+            if verbose:
+                print("Interpolating the numerical solution " +
+                      "associated with the key " +
+                      "'{}' ".format(solution_key) +
+                      "by default...")
+        elif solution_key not in self._solutions:
+            raise ValueError("no existing key " +
+                             "'{}' ".format(solution_key) +
+                             "referring to any numerical solution")
+
+        if method is None:
+            method = 'cubic spline'
+            if verbose:
+                print("Performing cubic spline interpolation by "
+                      "default...")
+
+        if interpolation_key is None:
+            interpolation_key = "{}-interp-".format(method)
+            interpolation_key += "{}".format(solution_key)
+            if verbose:
+                print("Resulting interpolation will be associated " +
+                      "with the key '{}' ".format(interpolation_key) +
+                      "by default.")
+
+        if method=='cubic spline':
+            self._interpolations[interpolation_key] = []
+            dim = self.codomain().dim()
+            for i in range(dim):
+                coordinate_curve = []
+                for point in self._solutions[solution_key]:
+                    coordinate_curve += [[point[0], point[i+1]]]
+                self._interpolations[interpolation_key]+=[Spline(coordinate_curve)]
+        else:
+            raise ValueError("no available method of interpolation " +
+                             "referred to as '{}'".format(method))
+
+        if verbose:
+            print("Interpolation completed and associated with the " +
+                  "key '{}' ".format(interpolation_key) +
+                  "(if this key already referred to a former " +
+                  "interpolation, such an interpolation was erased).")
+
+        return self._interpolations[interpolation_key]
+
+    def interpolation(self, interpolation_key=None, verbose=False):
+        r"""
+        Return the interpolation object associated with the given key.
+
+        INPUT:
+
+        - ``interpolation_key`` -- (default: ``None``) key which the
+          requested interpolation is associated to; a default
+          value is chosen if none is provided
+        - ``verbose`` -- (default: ``False``) prints information about
+          the interpolation object returned
+
+        OUTPUT:
+
+        - requested interpolation object
+
+        EXAMPLES:
+
+        Requesting an interpolation object previously computed::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q, L, T] = var('t B_0 m q L T')
+            sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c')
+            sage: sol = c.solve(method='rk4_maxima',
+            ....:        solution_key='sol_T1',
+            ....:        parameters_values={B_0:1, m:1, q:1, L:10, T:1})
+            sage: interp = c.interpolate(method='cubic spline',
+            ....:                         solution_key='sol_T1',
+            ....:                         interpolation_key='interp_T1')
+            sage: c.interpolation(interpolation_key='my interp')
+            Traceback (most recent call last):
+            ...
+            ValueError: no existing key 'my interp' referring to any
+             interpolation
+            sage: default_interp = c.interpolation(verbose=True)
+            Returning the interpolation associated with the key
+             'interp_T1' by default...
+            sage: default_interp == interp
+            True
+            sage: interp_mute = c.interpolation()
+            sage: interp_mute == interp
+            True
+
+        """
+
+        if interpolation_key==None:
+            if 'cubic spline' in self._interpolations:
+                interpolation_key = 'cubic spline'
+            else:
+                interpolation_key = self._interpolations.keys()[0]#will
+                # raise error if self._interpolations empty
+            if verbose:
+                print("Returning the interpolation associated with " +
+                      "the key '{}' ".format(interpolation_key) +
+                      "by default...")
+        elif interpolation_key not in self._interpolations:
+            raise ValueError("no existing key " +
+                             "'{}' ".format(interpolation_key) +
+                             "referring to any interpolation")
+
+        return self._interpolations[interpolation_key]
+
+    def __call__(self, t, interpolation_key=None,
+                 verbose=False):
+        r"""
+        Return the image of the curve for the given value of the curve
+        parameter, using the chosen interpolation.
+
+        INPUT:
+
+        - ``t'' -- curve parameter value at which the curve is evaluated
+        - ``interpolation_key`` -- (default: ``None``) key which the
+          interpolation requested to compute the point is associated to;
+          a default value is chosen if none is provided
+        - ``verbose`` -- (default: ``False``) prints information about
+          the interpolation used
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.point.ManifoldPoint` on a
+          manifold (codomain) with numerical coordinates
+
+        TESTS::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q, L, T] = var('t B_0 m q L T')
+            sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c')
+            sage: sol = c.solve(method='rk4_maxima',
+            ....:        solution_key='sol_T1',
+            ....:        parameters_values={B_0:1, m:1, q:1, L:10, T:1})
+            sage: interp = c.interpolate(method='cubic spline',
+            ....:                         solution_key='sol_T1',
+            ....:                         interpolation_key='interp_T1')
+            sage: c(1.1, interpolation_key='my interp')
+            Traceback (most recent call last):
+            ...
+            ValueError: no existing key 'my interp' referring to any
+             interpolation
+            sage: p = c(1.1, verbose=True)
+            Evaluating point coordinates from the interpolation
+             associated with the key 'interp_T1' by default...
+            sage: p.coordinates()     # abs tol 1e-12
+            (1.060743343394347, -0.2153835404373033, 1.1)
+
+        """
+
+        if interpolation_key is None:
+            if 'cubic spline' in self._interpolations:
+                interpolation_key = 'cubic spline'
+            else:
+                interpolation_key = self._interpolations.keys()[0]#will
+                # raise error if self._interpolations empty
+            if verbose:
+                print("Evaluating point coordinates from the " +
+                  "interpolation associated with the key " +
+                  "'{}' by default...".format(interpolation_key))
+        elif interpolation_key not in self._interpolations:
+            raise ValueError("no existing key " +
+                             "'{}' ".format(interpolation_key) +
+                             "referring to any interpolation")
+
+        interpolation = self._interpolations[interpolation_key]
+
+        if not isinstance(interpolation[0], Spline):
+            # partial test, in case future interpolation objects do not
+            # contain lists of instances of the Spline class
+            raise TypeError("unexpected type of interpolation object")
+
+        interpolated_coordinates = [coord_curve_spline(t)
+                                    for coord_curve_spline in interpolation]
+        return self.codomain().point(coords=interpolated_coordinates,
+                                                  chart=self._chart)
+
+    def tangent_vector_eval_at(self, t,
+                               interpolation_key=None, verbose=False):
+        r"""
+        Return the vector tangent to ``self`` at the given curve
+        parameter with components evaluated from the given
+        interpolation.
+
+        INPUT:
+
+        - ``t`` -- curve parameter value at which the tangent vector is
+          evaluated
+        - ``interpolation_key`` -- (default: ``None``) key which the
+          interpolation requested to compute the tangent vector is
+          associated to; a default value is chosen if none is provided
+        - ``verbose`` -- (default: ``False``) prints information about
+          the interpolation used
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.differentiable.tangent_vector.TangentVector`
+          tangent vector with numerical components
+
+        EXAMPLES:
+
+        Evaluating a vector tangent to the curve::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, B_0, m, q, L, T] = var('t B_0 m q L T')
+            sage: B = B_0*t/T*exp(-(x1^2 + x2^2)/L^2)
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [q*B/m*D[1], -q*B/m*D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,5), v, name='c')
+            sage: sol = c.solve(method='rk4_maxima',
+            ....:        solution_key='sol_T1',
+            ....:        parameters_values={B_0:1, m:1, q:1, L:10, T:1})
+            sage: interp = c.interpolate(method='cubic spline',
+            ....:                         solution_key='sol_T1',
+            ....:                         interpolation_key='interp_T1')
+            sage: tg_vec = c.tangent_vector_eval_at(1.22,
+            ....:                         interpolation_key='my interp')
+            Traceback (most recent call last):
+            ...
+            ValueError: no existing key 'my interp' referring to any
+             interpolation
+            sage: tg_vec = c.tangent_vector_eval_at(1.22, verbose=True)
+            Evaluating tangent vector components from the interpolation
+             associated with the key 'interp_T1' by default...
+            sage: tg_vec
+            Tangent vector at Point on the 3-dimensional differentiable
+             manifold M
+            sage: tg_vec[:]     # abs tol 1e-12
+            [0.7392639473853356, -0.6734182305341726, 1.0000000000000007]
+            sage: tg_vec_mute = c.tangent_vector_eval_at(1.22,
+            ....:                         interpolation_key='interp_T1')
+            sage: tg_vec_mute == tg_vec
+            True
+
+        """
+
+        if interpolation_key is None:
+            if 'cubic spline' in self._interpolations:
+                interpolation_key = 'cubic spline'
+            else:
+                # will raise error if self._interpolations empty
+                interpolation_key = self._interpolations.keys()[0]
+            if verbose:
+                print("Evaluating tangent vector components from the " +
+                      "interpolation associated with the key " +
+                      "'{}' by default...".format(interpolation_key))
+        elif interpolation_key not in self._interpolations:
+            raise ValueError("no existing key " +
+                             "'{}' ".format(interpolation_key) +
+                             "referring to any interpolation")
+
+        interpolation = self._interpolations[interpolation_key]
+
+        if not isinstance(interpolation[0], Spline):
+            # partial test, in case future interpolation objects do not
+            # contain lists of instances of the Spline class
+            raise TypeError("unexpected type of interpolation object")
+
+        interpolated_coordinates=[coordinate_curve_spline(t)
+                       for coordinate_curve_spline in interpolation]
+        M = self.codomain()
+        p = M.point(interpolated_coordinates, chart=self._chart, name=None)
+        Tp = M.tangent_space(p)
+
+        # by default, order=1 in method 'derivative' of a class Spline
+        evaluated_tgt_vec_comp = [coord_curve_spline.derivative(t)
+                                  for coord_curve_spline in interpolation]
+        basis = self._chart.frame().at(p)
+        return Tp(evaluated_tgt_vec_comp, basis=basis)
+
+    @options(thickness=1, plot_points=75, aspect_ratio='automatic',
+             plot_points_tangent=10, width_tangent=1, scale=1)
+    def plot_integrated(self, chart=None, ambient_coords=None,
+             mapping=None, prange=None, interpolation_key=None,
+             include_end_point=(True, True),
+             end_point_offset=(0.001, 0.001), verbose=False, color='red',
+             style='-', label_axes=True, display_tangent=False,
+             color_tangent='blue', **kwds):
+        r"""
+        Plot the 2D or 3D projection of ``self`` onto the space of the
+        chosen two or three ambient coordinates, based on the
+        interpolation of a numerical solution previously computed.
+
+        .. SEEALSO::
+
+            :class:`~sage.manifolds.differentiable.curve.DifferentiableCurve.plot`
+            for complete information about the input.
+
+        ADDITIONAL INPUT:
+
+        - ``interpolation_key`` -- (default: ``None``) key associated to
+          the interpolation object used for the plot; a default value
+          is chosen if none is provided
+        - ``verbose`` -- (default: ``False``) prints information about
+          the interpolation object used and the plotting in progress
+        - ``display_tangent`` -- (default: ``False``) determines whether
+          some tangent vectors should also be plotted
+        - ``color_tangent`` -- (default: ``blue``) color of the tangent
+          vectors when these are plotted
+        - ``plot_points_tangent`` -- (default: 10) number of tangent
+          vectors to display when these are plotted
+        - ``width_tangent`` -- (default: 1) sets the width of the arrows
+          representing the tangent vectors
+        - ``scale`` -- (default: 1) scale applied to the tangent vectors
+          before displaying them
+
+        EXAMPLES:
+
+        Trajectory of a particle of unit mass and unit charge in an
+        unit, axial, uniform, stationary magnetic field::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: var('t')
+            t
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [D[1], -D[0], 0]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,6), v, name='c')
+            sage: sol = c.solve()
+            sage: interp = c.interpolate()
+            sage: c_plot_2d = c.plot_integrated(ambient_coords=[x1, x2],
+            ....:                 thickness=2.5,
+            ....:                 display_tangent=True, plot_points=200,
+            ....:                 plot_points_tangent=10, scale=0.5,
+            ....:                 color='blue', color_tangent='red',
+            ....:                 verbose=True)
+            Plotting from the interpolation associated with the key
+             'cubic spline-interp-rk4_maxima' by default...
+            A tiny final offset equal to 0.000301507537688442 was
+             introduced for the last point in order to safely compute it
+             from the interpolation.
+            sage: c_plot_2d
+            Graphics object consisting of 11 graphics primitives
+
+        .. PLOT::
+
+            M = Manifold(3, 'M')
+            X = M.chart('x1 x2 x3'); x1, x2, x3 = X[:]
+            D = X.symbolic_velocities()
+            eqns = [D[1], -D[0], 0]
+            p = M.point((0,0,0), name='p')
+            Tp = M.tangent_space(p)
+            v = Tp((1,0,1))
+            t = var('t')
+            c = M.integrated_curve(eqns, D, (t, 0, 6), v, name='c')
+            sol = c.solve()
+            interp = c.interpolate()
+            c_plot_2d_1 = c.plot_integrated(ambient_coords=[x1, x2],
+                            thickness=2.5,
+                            display_tangent=True, plot_points=200,
+                            plot_points_tangent=10, scale=0.5,
+                            color='blue', color_tangent='red')
+            sphinx_plot(c_plot_2d_1)
+
+        """
+
+        from sage.manifolds.chart import RealChart
+
+        #
+        # Get the @options from kwds
+        #
+        thickness = kwds.pop('thickness')
+        plot_points = kwds.pop('plot_points')
+        aspect_ratio = kwds.pop('aspect_ratio')
+
+        #
+        # Interpolation to use
+        #
+        if interpolation_key is None:
+            if 'cubic spline' in self._interpolations:
+                interpolation_key = 'cubic spline'
+            else:
+                interpolation_key = self._interpolations.keys()[0]#will
+                # raise error if self._interpolations empty
+            if verbose:
+                print("Plotting from the interpolation associated " +
+                  "with the key '{}' ".format(interpolation_key) +
+                  "by default...")
+        elif interpolation_key not in self._interpolations:
+            raise ValueError("no existing key " +
+                             "'{}' ".format(interpolation_key) +
+                             "referring to any interpolation")
+
+        interpolation = self._interpolations[interpolation_key]
+
+        #
+        # The mapping, if present, and the chart with respect to which the curve
+        # is plotted
+        #
+        if mapping is None:
+            i0 = self.codomain().start_index()
+            if chart is None:
+                chart = self._chart
+            else:
+                if not isinstance(chart, RealChart):
+                    raise TypeError("{} is not a real " +
+                                    "chart".format(chart))
+                mapping = self.codomain().identity_map()
+        else:
+            i0 = mapping.codomain().start_index()
+            if chart is None:
+                chart = mapping.codomain().default_chart()
+            elif not isinstance(chart, RealChart):
+                raise TypeError("{} is not a real chart".format(chart))
+
+        #
+        # Coordinates of the above chart with respect to which the curve is
+        # plotted
+        #
+        if ambient_coords is None:
+            ambient_coords = chart[:]  # all chart coordinates are used
+        n_pc = len(ambient_coords)
+        if n_pc != 2 and n_pc !=3:
+            raise ValueError("the number of coordinates involved in " +
+                             "the plot must be either 2 or 3, " +
+                             "not {}".format(n_pc))
+
+        # From now on, 'pc' will denote coordinates in terms of which
+        # the curve is plotted (i.e. the "ambient coordinates"), while
+        # 'coord' will denote coordinates on self.domain(); of course,
+        # when, for instance, the mapping is the identity map, these may
+        # be the same.
+
+        # indices of plot coordinates
+        # will raise an error if ambient_coords are not associated with chart
+        ind_pc = [chart[:].index(pc) + i0 for pc in ambient_coords]
+
+        #
+        # Maximal parameter range for the plot of the chosen
+        # interpolation
+        #
+        # these two lines are the only general way to get the maximal
+        # parameter range since, at this point, there is no clue about
+        # the solution from which 'interpolation' was build, and it would
+        # be an obvious error to declare param_min=self.domain().lower_bound()
+        # for instance, since this might be an expression
+        param_min = interpolation[0][0][0]
+        param_max = interpolation[0][-1][0]
+
+        if prange is None:
+            prange = (param_min, param_max)
+        elif not isinstance(prange, (tuple, list)):
+            raise TypeError("{} is neither ".format(prange) +
+                            "a tuple nor a list")
+        elif len(prange) != 2:
+            raise ValueError("the argument prange must be a " +
+                             "tuple/list of 2 elements")
+        else:
+            p = prange #'p' declared only for the line below to be shorter
+            if p[0]<param_min or p[0]>param_max or p[1]<param_min or p[1]>param_max:
+                raise ValueError("parameter range should be a " +
+                                 "subinterval of the curve domain " +
+                                 "({})".format(self.domain()))
+
+        tmin = numerical_approx(prange[0])
+        tmax = numerical_approx(prange[1])
+
+        if not include_end_point[0]:
+            tmin += numerical_approx(end_point_offset[0])
+
+        if not include_end_point[1]:
+            tmax -= numerical_approx(end_point_offset[1])
+
+        if mapping is None:
+            if not isinstance(interpolation[0], Spline):
+                # partial test in case future interpolation objects do not
+                # contain lists of instances of the Spline class
+                raise TypeError("unexpected type of interpolation object")
+
+            #
+            # List of points for the plot curve
+            #
+            plot_curve = []
+            dt = (tmax - tmin) / (plot_points - 1)
+            t = tmin
+
+            for k in range(plot_points):
+                if k == 0 and t < param_min:
+                    # This might happen for the first point (i.e. k = 0)
+                    # when prange[0], and hence tmin, should equal param_min;
+                    # but mere numerical rounding coming from having taken
+                    # tmin = numerical_approx(prange[0]) might
+                    # raise errors from trying to evaluate the
+                    # interpolation at a time smaller than
+                    # self.domain.lower_bound(). Hence the line below
+                    # that adds 1% of the step to compute even more
+                    # safely the first point
+                    t = param_min + 0.01*dt
+                    if verbose:
+                        print("A tiny initial offset equal to " +
+                              "{} ".format(0.01*dt)+
+                              "was introduced for the first point "+
+                              "only, in order to safely compute " +
+                              "it from the interpolation.")
+
+                if k == plot_points-1 and t > param_max:
+                    # This might happen for the last point
+                    # (i.e. k = plot_points-1) when prange[1], and hence
+                    # tmax, should equal param_max; but mere numerical
+                    # rounding coming from having taken
+                    # tmax = numerical_approx(prange[1) might raise errors
+                    # from trying to evaluate the interpolation at a time
+                    # greater than self.domain.upper_bound().
+                    # Hence the line below that subtract 1% of the
+                    # step to compute even more safely the last point
+                    t = param_max - 0.01*dt
+                    if verbose:
+                        print("A tiny final offset equal to " +
+                              "{} ".format(0.01*dt)+
+                              "was introduced for the last point "+
+                              "in order to safely compute " +
+                              "it from the interpolation.")
+
+                plot_curve.append([interpolation[j-i0](t) for j in ind_pc])
+
+                if k == 0 and t > tmin:
+                    # in case an initial offset was earlier added to
+                    # 'tmin' in order to avoid errors, it is now needed
+                    # to cancel this offset for the next steps
+                    t = tmin
+                t += dt
+
+            if display_tangent:
+                from sage.plot.graphics import Graphics
+                from sage.plot.arrow import arrow2d
+                from sage.plot.plot3d.shapes import arrow3d
+
+                scale = kwds.pop('scale')
+                plot_points_tangent=kwds.pop('plot_points_tangent')
+                width_tangent = kwds.pop('width_tangent')
+
+                plot_vectors = Graphics()
+                dt = (tmax - tmin) / (plot_points_tangent - 1)
+                t = tmin
+
+                for k in range(plot_points_tangent):
+                    if k == 0 and t < param_min:
+                        # This might happen for the first point
+                        # (i.e. k = 0) when prange[0], and hence tmin
+                        # should equal param_min; but mere numerical
+                        # rounding coming from having taken
+                        # tmin = numerical_approx(prange[0]) might
+                        # raise errors from trying to evaluate the
+                        # interpolation at a time smaller than
+                        # self.domain.lower_bound().
+                        # Hence the line below that add 1% of the step
+                        # to compute even more safely the first point.
+                        t = param_min + 0.01*dt
+                        if verbose:
+                            print("A tiny initial offset equal to " +
+                                  "{} ".format(0.01*dt)+
+                                  "was introduced for the first point "+
+                                  "only, in order to safely compute " +
+                                  "it from the interpolation.")
+
+                    if k == plot_points_tangent - 1 and t > param_max:
+                        # This might happen for the last point
+                        # (i.e. k = plot_points_tangent-1) when
+                        # prange[1], and hence tmax, should equal
+                        # param_max; but mere numerical rounding coming from
+                        # having taken tmax = numerical_approx(prange[1)
+                        # might raise errors from trying to evaluate the
+                        # interpolation at a time greater than
+                        # self.domain.upper_bound(). Hence the line below
+                        # that subtracts 1% of the step to compute even
+                        # more safely the last point.
+                        t = param_max - 0.01*dt
+                        if verbose:
+                            print("A tiny final offset equal to " +
+                                  "{} ".format(0.01*dt)+
+                                  "was introduced for the last point "+
+                                  "in order to safely compute " +
+                                  "it from the interpolation.")
+
+                    # interpolated ambient coordinates:
+                    xp = [interpolation[j-i0](t) for j in ind_pc]
+
+                    # tangent vector ambiant components evaluated
+                    # from the interpolation:
+                    vec = [coordinate_curve_spline.derivative(t)
+                       for coordinate_curve_spline in interpolation]
+
+                    coord_tail = xp
+                    coord_head = [xp[j] + scale*vec[j]
+                                  for j in range(len(xp))]
+
+                    if coord_head != coord_tail:
+                        if n_pc == 2:
+                            plot_vectors += arrow2d(tailpoint=coord_tail,
+                                                    headpoint=coord_head,
+                                                    color=color_tangent,
+                                                    width=width_tangent)
+                        else:
+                            plot_vectors += arrow3d(coord_tail,
+                                                    coord_head,
+                                                    color=color_tangent,
+                                                    width=width_tangent)
+
+                    if k == 0 and t > tmin:
+                        # in case an initial offset was earlier added
+                        # to 'tmin' in order to avoid errors, it is now
+                        # needed to cancel this offset for the next steps
+                        t = tmin
+                    t += dt
+
+                return plot_vectors + DifferentiableCurve._graphics(self,
+                                         plot_curve, ambient_coords,
+                                         thickness=thickness,
+                                         aspect_ratio=aspect_ratio,
+                                         color=color,
+                                         style=style,
+                                         label_axes=label_axes)
+
+            return DifferentiableCurve._graphics(self, plot_curve,
+                             ambient_coords, thickness=thickness,
+                             aspect_ratio=aspect_ratio, color=color,
+                             style=style, label_axes=label_axes)
+        else:
+            #
+            # The coordinate expressions of the mapping and the
+            # coordinates involved
+            #
+            for chart_pair in mapping._coord_expression:
+                subs = (chart_pair[0]._subcharts, chart_pair[1]._subcharts)
+                # 'subs' declared only for the line below to be shorter
+                if self._chart in subs[0] and chart in subs[1]:
+                    transf = {}
+                    required_coords = set()
+                    for pc in ambient_coords:
+                        j = chart[:].index(pc)
+                        AUX = mapping._coord_expression[chart_pair]
+                        # 'AUX' used only for the lines of source code
+                        # to be shorter
+                        transf[pc] = AUX.expr()[j]
+                        AUX2 = transf[pc].variables() # idem
+                        required_coords=required_coords.union(AUX2)
+                    break
+            else:
+                raise ValueError("no expression has been found for " +
+                                 "{} in terms of {}".format(self,chart))
+
+            if not isinstance(interpolation[0], Spline):
+                # partial test, in case future interpolation objects do not
+                # contain lists of instances of the Spline class
+                raise TypeError("unexpected type of interpolation object")
+
+            #
+            # List of points for the plot curve
+            #
+            plot_curve = []
+            dt = (tmax - tmin) / (plot_points - 1)
+            t = tmin
+            required_coords_values = {}
+
+            for k in range(plot_points):
+                if k == 0 and t < param_min:
+                    # This might happen for the first point (i.e. k = 0)
+                    # when prange[0], and hence tmin, should equal param_min;
+                    # but mere numerical rounding coming from having taken
+                    # tmin = numerical_approx(prange[0]) might
+                    # raise errors from trying to evaluate the
+                    # interpolation at a time smaller than
+                    # self.domain.lower_bound(). Hence the line below that adds
+                    # 1% of the step to compute even more safely the first point
+                    t = param_min + 0.01*dt
+                    if verbose:
+                        print("A tiny initial offset equal to " +
+                              "{} ".format(0.01*dt)+
+                              "was introduced for the first point "+
+                              "only, in order to safely compute " +
+                              "it from the interpolation.")
+
+                if k == plot_points - 1 and t > param_max:
+                    # This might happen for the last point (i.e. k = plot_points-1)
+                    # when prange[1], and hence tmax, should equal
+                    # param_max; but mere numerical rounding coming from
+                    # having taken tmax = numerical_approx(prange[1)
+                    # might raise errors from trying to evaluate the
+                    # interpolation at a time greater than
+                    # self.domain.upper_bound(). Hence the line below that
+                    # subtracts 1% of the step to compute even more safely
+                    # the last point.
+                    t = param_max - 0.01*dt
+                    if verbose:
+                        print("A tiny final offset equal to " +
+                              "{} ".format(0.01*dt)+
+                              "was introduced for the last point "+
+                              "in order to safely compute " +
+                              "it from the interpolation.")
+
+                for coord in required_coords:
+                    i = self._chart[:].index(coord)
+                    required_coords_values[coord] = interpolation[i](t)
+
+                xp = []
+                for j in ind_pc:
+                    pc = chart[j]
+                    AUX = transf[pc]
+                    AUX = AUX.substitute(required_coords_values)
+                    # 'AUX' only used for the lines of source code
+                    #  to be shorter
+                    xp += [numerical_approx(AUX)]
+
+                plot_curve.append(xp)
+
+                if k==0 and t > tmin:
+                    # in case an initial offset was earlier added to
+                    # 'tmin' in order to avoid errors, it is now needed
+                    # to cancel this offset for the next steps
+                    t=tmin
+
+                t += dt
+
+            if display_tangent:
+                from sage.plot.graphics import Graphics
+                from sage.plot.arrow import arrow2d
+                from sage.plot.plot3d.shapes import arrow3d
+
+                scale = kwds.pop('scale')
+                plot_points_tangent = kwds.pop('plot_points_tangent')
+                width_tangent = kwds.pop('width_tangent')
+
+                plot_vectors = Graphics()
+                dt = (tmax - tmin) / (plot_points_tangent - 1)
+                t = tmin
+                Dcoord_Dt = {}
+
+                Dpc_Dcoord = {}
+                for pc in ambient_coords:
+                    Dpc_Dcoord[pc] = {}
+                    for coord in transf[pc].variables():
+                        Dpc_Dcoord[pc][coord] = transf[pc].derivative(coord)
+
+                for k in range(plot_points_tangent):
+                    if k == 0 and t < param_min:
+                        # This might happen for the first point (i.e. k = 0)
+                        # when prange[0], and hence tmin, should equal param_min;
+                        # but mere numerical rounding coming from having taken
+                        # tmin = numerical_approx(prange[0]) might
+                        # raise errors from trying to evaluate the
+                        # interpolation at a time smaller than
+                        # self.domain.lower_bound(). Hence the line below
+                        # that adds 1% of the step to compute even more
+                        # safely the first point
+                        t = param_min + 0.01*dt
+                        if verbose:
+                            print("A tiny initial offset equal to " +
+                                  "{} ".format(0.01*dt)+
+                                  "was introduced for the first point "+
+                                  "only, in order to safely compute " +
+                                  "it from the interpolation.")
+
+                    if k == plot_points_tangent - 1 and t > param_max:
+                        # This might happen for the last point
+                        # (i.e. k = plot_points_tangent-1) when
+                        # when prange[1], and hence tmax, should equal
+                        # param_max; but mere numerical rounding coming from
+                        # having taken tmax = numerical_approx(prange[1)
+                        # might raise errors from trying to evaluate the
+                        # interpolation at a time greater than
+                        # self.domain.upper_bound(). Hence the line below
+                        # that subtracts 1% of the step to compute even
+                        # more safely the last point
+                        t = param_max - 0.01*dt
+                        if verbose:
+                            print("A tiny final offset equal to " +
+                                  "{} ".format(0.01*dt)+
+                                  "was introduced for the last point "+
+                                  "in order to safely compute " +
+                                  "it from the interpolation.")
+
+                    for coord in required_coords:
+                        i = self._chart[:].index(coord)
+                        AUX = interpolation[i] # 'AUX' only used
+                        # for the lines below to be shorter
+                        required_coords_values[coord] = AUX(t)
+                        Dcoord_Dt[coord] = AUX.derivative(t)
+
+                    xp = []
+                    pushed_vec = []
+                    for j in ind_pc:
+                        pc = chart[j]
+                        AUX = transf[pc]
+                        AUX = AUX.substitute(required_coords_values)
+                        # 'AUX' only used for the lines of code to
+                        # be shorter
+                        xp+=[numerical_approx(AUX)]
+
+                        pushed_comp = 0
+                        for coord in transf[pc].variables():
+                            D = Dpc_Dcoord[pc][coord]
+                            D = D.substitute(required_coords_values)
+                            D=numerical_approx(D)
+                            pushed_comp += Dcoord_Dt[coord] * D
+
+                        pushed_vec += [pushed_comp]
+
+                    coord_tail = xp
+                    coord_head = [val + scale*pushed_vec[j]
+                                  for j, val in enumerate(xp)]
+
+                    if coord_head != coord_tail:
+                        if n_pc == 2:
+                            plot_vectors += arrow2d(tailpoint=coord_tail,
+                                                    headpoint=coord_head,
+                                                    color=color_tangent,
+                                                    width=width_tangent)
+                        else:
+                            plot_vectors += arrow3d(coord_tail,
+                                                    coord_head,
+                                                    color=color_tangent,
+                                                    width=width_tangent)
+
+                    if k == 0 and t > tmin:
+                        # in case an initial offset was earlier added to
+                        # 'tmin' in order to avoid errors, it is now needed
+                        # to cancel this offset for the next steps
+                        t=tmin
+
+                    t += dt
+
+                return plot_vectors + DifferentiableCurve._graphics(self,
+                                         plot_curve, ambient_coords,
+                                         thickness=thickness,
+                                         aspect_ratio=aspect_ratio,
+                                         color=color,
+                                         style=style,
+                                         label_axes=label_axes)
+
+            return DifferentiableCurve._graphics(self, plot_curve,
+                             ambient_coords, thickness=thickness,
+                             aspect_ratio=aspect_ratio, color=color,
+                             style=style, label_axes=label_axes)
+
+class IntegratedAutoparallelCurve(IntegratedCurve):
+    r"""
+    Autoparallel curve on the manifold with respect to a given
+    affine connection.
+
+    INPUT:
+
+    - ``parent`` --
+      :class:`~sage.manifolds.differentiable.manifold_homset.IntegratedAutoparallelCurveSet`
+      the set of curves `\mathrm{Hom_{autoparallel}}(I, M)` to which the
+      curve belongs
+    - ``affine_connection`` --
+      :class:`~sage.manifolds.differentiable.affine_connection.AffineConnection`
+      affine connection with respect to which the curve is autoparallel
+    - ``curve_parameter`` -- symbolic expression to be used as the
+      parameter of the curve (the equations defining an instance of
+      IntegratedAutoparallelCurve are such that ``t`` will actually be
+      an affine parameter of the curve)
+    - ``initial_tangent_vector`` --
+      :class:`~sage.manifolds.differentiable.tangent_vector.TangentVector`
+      initial tangent vector of the curve
+    - ``chart`` -- (default: ``None``) chart on the manifold in terms of
+      which the equations are expressed; if ``None`` the default chart
+      of the manifold is assumed
+    - ``name`` -- (default: ``None``) string; symbol given to the curve
+    - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to
+      denote the curve; if none is provided, ``name`` will be used
+
+    EXAMPLES:
+
+    Autoparallel curves associated with the Mercator projection of the
+    unit 2-sphere `\mathbb{S}^{2}`.
+
+    .. SEEALSO::
+
+        https://idontgetoutmuch.wordpress.com/2016/11/24/mercator-a-connection-with-torsion/
+        for more details about Mercator projection.
+
+    On the Mercator projection, the lines of longitude all appear
+    vertical and then all parallel with respect to each other.
+    Likewise, all the lines of latitude appear horizontal and parallel
+    with respect to each other.
+    These curves may be recovered as autoparallel curves of a certain
+    connection `\nabla` to be made explicit.
+
+    Start with declaring the standard polar coordinates
+    `(\theta, \phi)` on `\mathbb{S}^{2}` and the
+    corresponding coordinate frame `(e_{\theta}, e_{\phi})`::
+
+        sage: S2 = Manifold(2, 'S^2', start_index=1)
+        sage: polar.<th,ph>=S2.chart()
+        sage: epolar = polar.frame()
+
+    Normalizing `e_{\phi}` provides an orthonormal basis::
+
+        sage: ch_basis = S2.automorphism_field()
+        sage: ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
+        sage: epolar_ON = epolar.new_frame(ch_basis,'epolar_ON')
+
+    Denote `(\hat{e}_{\theta}, \hat{e}_{\phi})` such an orthonormal frame
+    field. In any point, the vector field `\hat{e}_{\theta}` is
+    normalized and tangent to the line of longitude through the point.
+    Likewise, `\hat{e}_{\phi}` is normalized and tangent to the
+    line of latitude.
+
+    Now, set an affine connection with respect to such fields that are
+    parallely transported in all directions, that is:
+    `\nabla \hat{e}_{\theta} = \nabla \hat{e}_{\phi} = 0`.
+    This is equivalent to setting all the connection coefficients to
+    zero with respect to this frame::
+
+        sage: nab = S2.affine_connection('nab')
+        sage: nab.set_coef(frame=epolar_ON)[:]
+        [[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
+
+    This connection is such that two vectors are parallel if their
+    angles to a given meridian are the same.
+    Check that this connection is compatible with the Euclidean
+    metric tensor `g` induced on `\mathbb{S}^{2}`::
+
+        sage: g = S2.metric('g')
+        sage: g[1,1], g[2,2] = 1, (sin(th))^2
+        sage: nab(g)[:]
+        [[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
+
+    Yet, this connection is not the Levi-Civita connection, which
+    implies that it has non-vanishing torsion::
+
+        sage: nab.torsion()[:]
+        [[[0, 0], [0, 0]], [[0, cos(th)/sin(th)], [-cos(th)/sin(th), 0]]]
+
+    Set generic initial conditions for the autoparallel curves to
+    compute::
+
+        sage: [th0, ph0, v_th0, v_ph0] = var('th0 ph0 v_th0 v_ph0')
+        sage: p = S2.point((th0, ph0), name='p')
+        sage: Tp = S2.tangent_space(p)
+        sage: v = Tp((v_th0, v_ph0), basis=epolar_ON.at(p))
+
+    Note here that the components ``(v_th0, v_ph0)`` of the initial
+    tangent vector ``v`` refer to the basis
+    ``epolar_ON`` `= (\hat{e}_{\theta}, \hat{e}_{\phi})`
+    and not the coordinate basis ``epolar`` `= (e_{\theta}, e_{\phi})`.
+    This is merely to help picture the aspect of the tangent vector in
+    the usual embedding of `\mathbb{S}^{2}` in
+    `\mathbb{R}^{3}` thanks to using an orthonormal frame,
+    since providing the components with respect to the coordinate basis
+    would require mutliplying the second component (i.e. the `\phi`
+    component) in order to picture the vector in the same way.
+    This subtlety will need to be taken into account later when the
+    numerical curve will be compared to the analytical solution.
+
+    Now, declare the corresponding integrated autoparallel curve and display
+    the differential system it satisfies::
+
+        sage: [t, tmin, tmax] = var('t tmin tmax')
+        sage: c = S2.integrated_autoparallel_curve(nab, (t, tmin, tmax),
+        ....:                                  v, chart=polar, name='c')
+        sage: sys = c.system(verbose=True)
+        Autoparallel curve c in the 2-dimensional differentiable
+         manifold S^2 equipped with Affine connection nab on the
+         2-dimensional differentiable manifold S^2, and integrated over
+         the Real interval (tmin, tmax) as a solution to the following
+         equations, written with respect to Chart (S^2, (th, ph)):
+        <BLANKLINE>
+        Initial point: Point p on the 2-dimensional differentiable
+         manifold S^2 with coordinates [th0, ph0] with respect to
+         Chart (S^2, (th, ph))
+        Initial tangent vector: Tangent vector at Point p on the
+         2-dimensional differentiable manifold S^2 with
+         components [v_th0, v_ph0/sin(th0)] with respect to Chart (S^2, (th, ph))
+        <BLANKLINE>
+        d(th)/dt = Dth
+        d(ph)/dt = Dph
+        d(Dth)/dt = 0
+        d(Dph)/dt = -Dph*Dth*cos(th)/sin(th)
+        <BLANKLINE>
+
+    Set a dictionnary providing the parameter range and the initial
+    conditions for a line of latitude and a line of longitude::
+
+        sage: dict_params={'latit':{tmin:0,tmax:3,th0:pi/4,ph0:0.1,v_th0:0,v_ph0:1},
+        ....:   'longi':{tmin:0,tmax:3,th0:0.1,ph0:0.1,v_th0:1,v_ph0:0}}
+
+    Declare the Mercator coordinates `(\xi, \zeta)` and the
+    corresponding coordinate change from the polar coordinates::
+
+        sage: mercator.<xi,ze> = S2.chart(r'xi:(-oo,oo):\xi ze:(0,2*pi):\zeta')
+        sage: polar.transition_map(mercator, (log(tan(th/2)), ph))
+        Change of coordinates from Chart (S^2, (th, ph)) to Chart
+         (S^2, (xi, ze))
+
+    Ask for the identity map in terms of these charts in order to add
+    this coordinate change to its dictionnary of expressions. This is
+    required to plot the curve with respect to the Mercator chart::
+
+        sage: identity = S2.identity_map()
+        sage: identity.coord_functions(polar, mercator)
+        Coordinate functions (log(sin(1/2*th)/cos(1/2*th)), ph) on the
+         Chart (S^2, (th, ph))
+
+    Solve, interpolate and prepare the plot for the solutions
+    corresponding to the two initial conditions previously set::
+
+        sage: graph2D_mercator = Graphics()
+        sage: for key in dict_params:
+        ....:     sol = c.solve(solution_key='sol-'+key,
+        ....:                        parameters_values=dict_params[key])
+        ....:     interp = c.interpolate(solution_key='sol-'+key,
+        ....:                           interpolation_key='interp-'+key)
+        ....:     graph2D_mercator+=c.plot_integrated(interpolation_key='interp-'+key,
+        ....:                               chart=mercator, thickness=2)
+
+    Prepare a grid of Mercator coordinates lines, and plot the curves
+    over it::
+
+        sage: graph2D_mercator_coords=mercator.plot(chart=mercator,
+        ....:                            number_values=8,color='yellow')
+        sage: graph2D_mercator + graph2D_mercator_coords
+        Graphics object consisting of 18 graphics primitives
+
+    .. PLOT::
+
+        S2 = Manifold(2, 'S^2', start_index=1)
+        polar = S2.chart('th ph'); th, ph = polar[:]
+        epolar = polar.frame()
+        ch_basis = S2.automorphism_field()
+        ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
+        epolar_ON = epolar.new_frame(ch_basis, 'epolar_ON')
+        nab = S2.affine_connection('nab')
+        _ = nab.set_coef(frame=epolar_ON)
+        t,tmin,tmax,th0,ph0,v_th0,v_ph0 = var('t tmin tmax th0 ph0 v_th0 v_ph0')
+        p = S2.point((th0, ph0), name='p')
+        Tp = S2.tangent_space(p)
+        v = Tp((v_th0, v_ph0), basis=epolar_ON.at(p))
+        c = S2.integrated_autoparallel_curve(nab, (t, tmin, tmax), v,
+                                                  chart=polar, name='c')
+        dict_params={'latit':{tmin:0,tmax:3,th0:pi/4,ph0:0.1,v_th0:0,v_ph0:1},
+                'longi':{tmin:0,tmax:3,th0:0.1,ph0:0.1,v_th0:1,v_ph0:0}}
+        mercator = S2.chart(r'xi:(-oo,oo):\xi ze:(0,2*pi):\zeta')
+        xi,ze = var('xi ze')
+        _ = polar.transition_map(mercator, (log(tan(th/2)), ph))
+        identity = S2.identity_map()
+        identity.coord_functions(polar, mercator)
+        graph2D_mercator = Graphics()
+        for key in dict_params:
+            sol = c.solve(solution_key='sol-'+key,
+                          parameters_values=dict_params[key])
+            interp = c.interpolate(solution_key='sol-'+key,
+                                   interpolation_key='interp-'+key)
+            graph2D_mercator += c.plot_integrated(interpolation_key='interp-'+key,
+                                            chart=mercator, thickness=2)
+        graph2D_mercator_coords = mercator.plot(chart=mercator,
+                                                number_values=8, color='yellow')
+        sphinx_plot(graph2D_mercator + graph2D_mercator_coords)
+
+    The resulting curves are horizontal and vertical as expected.
+    It is easier to check that these are latitude and longitude lines
+    respectively when plotting them on `\mathbb{S}^{2}`.
+    To do so, use `\mathbb{R}^{3}` as the codomain of the standard
+    map embedding `(\mathbb{S}^{2}, (\theta, \phi))` in the
+    3-dimensional Euclidean space::
+
+        sage: R3 = Manifold(3, 'R3', start_index=1)
+        sage: cart.<X,Y,Z> = R3.chart()
+        sage: euclid_embedding = S2.diff_map(R3,
+        ....:  {(polar, cart):[sin(th)*cos(ph),sin(th)*sin(ph),cos(th)]})
+
+    Plot the resulting curves on the grid of polar coordinates lines on
+    `\mathbb{S}^{2}`::
+
+        sage: graph3D_embedded_curves = Graphics()
+        sage: for key in dict_params:
+        ....:     graph3D_embedded_curves += c.plot_integrated(interpolation_key='interp-'+key,
+        ....:            mapping=euclid_embedding, thickness=5,
+        ....:            display_tangent=True, scale=0.4, width_tangent=0.5)
+        sage: graph3D_embedded_polar_coords = polar.plot(chart=cart,
+        ....:                          mapping=euclid_embedding,
+        ....:                          number_values=15, color='yellow')
+        sage: graph3D_embedded_curves + graph3D_embedded_polar_coords
+        Graphics3d Object
+
+    .. PLOT::
+
+        S2 = Manifold(2, 'S^2', start_index=1)
+        polar = S2.chart('th ph'); th, ph = polar[:]
+        epolar = polar.frame()
+        ch_basis = S2.automorphism_field()
+        ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
+        epolar_ON = epolar.new_frame(ch_basis, 'epolar_ON')
+        nab = S2.affine_connection('nab')
+        _ = nab.set_coef(frame=epolar_ON)
+        t,tmin,tmax,th0,ph0,v_th0,v_ph0 = var('t tmin tmax th0 ph0 v_th0 v_ph0')
+        p = S2.point((th0, ph0), name='p')
+        Tp = S2.tangent_space(p)
+        v = Tp((v_th0, v_ph0), basis=epolar_ON.at(p))
+        c = S2.integrated_autoparallel_curve(nab, (t, tmin, tmax), v,
+                                             chart=polar, name='c')
+        dict_params = {'latit':{tmin:0,tmax:3,th0:pi/4,ph0:0.1,v_th0:0,v_ph0:1},
+                       'longi':{tmin:0,tmax:3,th0:0.1,ph0:0.1,v_th0:1,v_ph0:0}}
+        R3 = Manifold(3, 'R3', start_index=1)
+        cart = R3.chart('X Y Z'); X, Y, Z = cart[:]
+        euclid_embedding = S2.diff_map(R3,
+              {(polar, cart): [sin(th)*cos(ph),sin(th)*sin(ph),cos(th)]})
+        graph3D_embedded_curves = Graphics()
+        for key in dict_params:
+            sol = c.solve(solution_key='sol-'+key,
+                          parameters_values=dict_params[key])
+            interp = c.interpolate(solution_key='sol-'+key,
+                                   interpolation_key='interp-'+key)
+            graph3D_embedded_curves += c.plot_integrated(interpolation_key='interp-'+key,
+                     mapping=euclid_embedding, thickness=5,
+                     display_tangent=True, scale=0.4, width_tangent=0.5)
+        graph3D_embedded_polar_coords = polar.plot(chart=cart,
+                                       mapping=euclid_embedding,
+                                       number_values=15, color='yellow')
+        graph = graph3D_embedded_curves+graph3D_embedded_polar_coords
+        sphinx_plot(graph)
+
+    Finally, one may plot a general autoparallel curve with respect to
+    `\nabla` that is neither a line of latitude or longitude.
+    The vectors tangent to such a curve make an angle different from 0
+    or `\pi/2` with the lines of latitude and longitude.
+    Then, compute a curve such that both components of its initial
+    tangent vectors are non zero::
+
+        sage: sol = c.solve(solution_key='sol-angle',
+        ....:  parameters_values={tmin:0,tmax:2,th0:pi/4,ph0:0.1,v_th0:1,v_ph0:8})
+        sage: interp = c.interpolate(solution_key='sol-angle',
+        ....:                          interpolation_key='interp-angle')
+
+    Plot the resulting curve in the Mercator plane.
+    This generates a straight line, as expected::
+
+        sage: c.plot_integrated(interpolation_key='interp-angle',
+        ....:         chart=mercator, thickness=1, display_tangent=True,
+        ....:         scale=0.2, width_tangent=0.2)
+        Graphics object consisting of 11 graphics primitives
+
+    .. PLOT::
+
+        S2 = Manifold(2, 'S^2', start_index=1)
+        polar = S2.chart('th ph'); th, ph = polar[:]
+        epolar = polar.frame()
+        ch_basis = S2.automorphism_field()
+        ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
+        epolar_ON = epolar.new_frame(ch_basis, 'epolar_ON')
+        nab = S2.affine_connection('nab')
+        _ = nab.set_coef(frame=epolar_ON)
+        t,tmin,tmax,th0,ph0,v_th0,v_ph0 = var('t tmin tmax th0 ph0 v_th0 v_ph0')
+        p = S2.point((th0, ph0), name='p')
+        Tp = S2.tangent_space(p)
+        v = Tp((v_th0, v_ph0), basis=epolar_ON.at(p))
+        c = S2.integrated_autoparallel_curve(nab, (t, tmin, tmax), v,
+                                                  chart=polar, name='c')
+        mercator = S2.chart(r'xi:(-oo,oo):\xi ze:(0,2*pi):\zeta')
+        xi, ze = mercator[:]
+        trans_map = polar.transition_map(mercator, (log(tan(th/2)), ph))
+        identity = S2.identity_map()
+        _ = identity.coord_functions(polar, mercator)
+        sol = c.solve(solution_key='sol-angle',
+            parameters_values={tmin:0,tmax:2,th0:pi/4,ph0:0.1,v_th0:1,v_ph0:8})
+        interp = c.interpolate(solution_key='sol-angle',
+                               interpolation_key='interp-angle')
+        graph2D_mercator_angle_curve=c.plot_integrated(
+                      interpolation_key='interp-angle',
+                      chart=mercator, thickness=1, display_tangent=True,
+                      scale=0.2, width_tangent=0.2)
+        sphinx_plot(graph2D_mercator_angle_curve)
+
+    One may eventually plot such a curve on `\mathbb{S}^{2}`::
+
+        sage: graph3D_embedded_angle_curve=c.plot_integrated(interpolation_key='interp-angle',
+        ....:        mapping=euclid_embedding, thickness=5,
+        ....:        display_tangent=True, scale=0.1, width_tangent=0.5)
+        sage: graph3D_embedded_angle_curve + graph3D_embedded_polar_coords
+        Graphics3d Object
+
+    .. PLOT::
+
+        S2 = Manifold(2, 'S^2', start_index=1)
+        polar = S2.chart('th ph'); th, ph = polar[:]
+        epolar = polar.frame()
+        ch_basis = S2.automorphism_field()
+        ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
+        epolar_ON = epolar.new_frame(ch_basis, 'epolar_ON')
+        nab = S2.affine_connection('nab')
+        _ = nab.set_coef(frame=epolar_ON)
+        t,tmin,tmax,th0,ph0,v_th0,v_ph0 = var('t tmin tmax th0 ph0 v_th0 v_ph0')
+        p = S2.point((th0, ph0), name='p')
+        Tp = S2.tangent_space(p)
+        v = Tp((v_th0, v_ph0), basis=epolar_ON.at(p))
+        c = S2.integrated_autoparallel_curve(nab, (t, tmin, tmax), v,
+                                                  chart=polar, name='c')
+        R3 = Manifold(3, 'R3', start_index=1)
+        cart = R3.chart('X Y Z')
+        euclid_embedding = S2.diff_map(R3,
+              {(polar, cart):[sin(th)*cos(ph),sin(th)*sin(ph),cos(th)]})
+        sol = c.solve(solution_key='sol-angle',
+            parameters_values={tmin:0,tmax:2,th0:pi/4,ph0:0.1,v_th0:1,v_ph0:8})
+        interp = c.interpolate(solution_key='sol-angle',
+                               interpolation_key='interp-angle')
+        graph3D_embedded_angle_curve = c.plot_integrated(interpolation_key='interp-angle',
+            mapping=euclid_embedding, thickness=5, display_tangent=True,
+            scale=0.1, width_tangent=0.5)
+        graph3D_embedded_polar_coords = polar.plot(chart=cart,
+             mapping=euclid_embedding, number_values=15, color='yellow')
+        graph = graph3D_embedded_angle_curve + graph3D_embedded_polar_coords
+        sphinx_plot(graph)
+
+    All the curves presented are loxodromes, and the differential system
+    defining them (displayed above) may be solved analytically,
+    providing the following expressions:
+
+    .. MATH::
+
+        \begin{aligned}
+        \theta(t) &= \theta_{0} + \dot{\theta}_{0} (t - t_{0}),      \\
+        \phi(t) &= \phi_{0} - \frac{1}{\tan \alpha} \left(
+        \ln \tan \frac{\theta_{0} + \dot{\theta}_{0} (t - t_{0})}{2} -
+        \ln \tan \frac{\theta_{0}}{2} \right),
+        \end{aligned}
+
+    where `\alpha` is the angle between the curve and any latitude
+    line it crosses; then, one finds
+    `\tan \alpha = - \dot{\theta}_{0} / (\dot{\phi}_{0} \sin \theta_{0})`
+    (then `\tan \alpha \leq 0` when the initial tangent vector
+    points towards the southeast).
+
+    In order to use these expressions to compare with the result
+    provided by the numerical integration, remember that the components
+    ``(v_th0, v_ph0)`` of the initial
+    tangent vector ``v`` refer to the basis
+    ``epolar_ON`` `= (\hat{e}_{\theta}, \hat{e}_{\phi})` and not the
+    coordinate basis
+    ``epolar`` `= (e_{\theta}, e_{\phi})`.
+    Therefore, the following relations hold:
+    ``v_ph0`` `= \dot{\phi}_{0} \sin \theta_{0}` (and not merely
+    `\dot{\phi}_{0}`), while ``v_th0`` clearly is `\dot{\theta}_{0}`.
+
+    With this in mind, plot an analytical curve to compare with a
+    numerical solution::
+
+        sage: graph2D_mercator_angle_curve=c.plot_integrated(interpolation_key='interp-angle',
+        ....:                               chart=mercator, thickness=1)
+        sage: expr_ph = ph0+v_ph0/v_th0*(ln(tan((v_th0*t+th0)/2))-ln(tan(th0/2)))
+        sage: c_loxo = S2.curve({polar:[th0+v_th0*t, expr_ph]}, (t,0,2),
+        ....:                                             name='c_loxo')
+
+    Ask for the expression of the loxodrome in terms of the Mercator
+    chart in order to add it to its dictionnary of expressions.
+    It is a particularly long expression, and there is no particular
+    need to diplay it, which is why it may simply be affected to an
+    arbitrary variable ``expr_mercator``, which will never be used
+    again.
+    But adding the expression to the dictionnary is required to plot the
+    curve with respect to the Mercator chart::
+
+        sage: expr_mercator = c_loxo.expression(chart2=mercator)
+
+    Plot the curves (for clarity, set a 2 degrees shift in the initial
+    value of `\theta_{0}` so that the curves do not overlap)::
+
+        sage: graph2D_mercator_loxo = c_loxo.plot(chart=mercator,
+        ....:  parameters={th0:pi/4+2*pi/180, ph0:0.1, v_th0:1, v_ph0:8},
+        ....:  thickness=1, color='blue')
+        sage: graph2D_mercator_angle_curve + graph2D_mercator_loxo
+        Graphics object consisting of 2 graphics primitives
+
+    .. PLOT::
+
+        S2 = Manifold(2, 'S^2', start_index=1)
+        polar = S2.chart('th ph'); th, ph = polar[:]
+        epolar = polar.frame()
+        ch_basis = S2.automorphism_field()
+        ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
+        epolar_ON = epolar.new_frame(ch_basis, 'epolar_ON')
+        nab = S2.affine_connection('nab')
+        _ = nab.set_coef(frame=epolar_ON)
+        t, tmin, tmax, th0, ph0 = var('t tmin tmax th0 ph0')
+        v_th0, v_ph0, alpha = var('v_th0 v_ph0 alpha')
+        p = S2.point((th0, ph0), name='p')
+        Tp = S2.tangent_space(p)
+        v = Tp((v_th0, v_ph0), basis=epolar_ON.at(p))
+        c = S2.integrated_autoparallel_curve(nab, (t, tmin, tmax), v,
+                                                  chart=polar, name='c')
+        mercator = S2.chart(r'xi:(-oo,oo):\xi ze:(0,2*pi):\zeta')
+        xi, ze = mercator[:]
+        trans_map = polar.transition_map(mercator, (log(tan(th/2)), ph))
+        identity = S2.identity_map()
+        _ = identity.coord_functions(polar, mercator)
+        sol = c.solve(solution_key='sol-angle',
+            parameters_values={tmin:0,tmax:2,th0:pi/4,ph0:0.1,v_th0:1,v_ph0:8})
+        interp = c.interpolate(solution_key='sol-angle',
+                               interpolation_key='interp-angle')
+        graph2D_mercator_angle_curve = c.plot_integrated(interpolation_key='interp-angle',
+                                                         chart=mercator, thickness=1)
+        expr_ph = ph0+v_ph0/v_th0*(ln(tan((v_th0*t+th0)/2))-ln(tan(th0/2)))
+        c_loxo = S2.curve({polar: [th0+v_th0*t, expr_ph]}, (t,0,2), name='c_loxo')
+        expr = c_loxo.expression(chart2=mercator)
+        graph2D_mercator_loxo = c_loxo.plot(chart=mercator,
+              parameters={th0:pi/4+2*pi/180, ph0:0.1, v_th0:1, v_ph0:8},
+              thickness=1, color='blue')
+        sphinx_plot(graph2D_mercator_angle_curve+graph2D_mercator_loxo)
+
+    Both curves do have the same aspect.
+    One may eventually compare these curves on `\mathbb{S}^{2}`::
+
+        sage: graph3D_embedded_angle_curve=c.plot_integrated(interpolation_key='interp-angle',
+        ....:                     mapping=euclid_embedding, thickness=3)
+        sage: graph3D_embedded_loxo = c_loxo.plot(mapping=euclid_embedding,
+        ....:  parameters={th0:pi/4+2*pi/180, ph0:0.1, v_th0:1, v_ph0:8},
+        ....:  thickness=3, color = 'blue')
+        sage: (graph3D_embedded_angle_curve + graph3D_embedded_loxo
+        ....:  + graph3D_embedded_polar_coords)
+        Graphics3d Object
+
+    .. PLOT::
+
+        S2 = Manifold(2, 'S^2', start_index=1)
+        polar = S2.chart('th ph'); th, ph = polar[:]
+        epolar = polar.frame()
+        ch_basis = S2.automorphism_field()
+        ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
+        epolar_ON = epolar.new_frame(ch_basis, 'epolar_ON')
+        nab = S2.affine_connection('nab')
+        _ = nab.set_coef(frame=epolar_ON)
+        t, tmin, tmax, th0, ph0 = var('t tmin tmax th0 ph0')
+        v_th0, v_ph0, alpha = var('v_th0 v_ph0 alpha')
+        p = S2.point((th0, ph0), name='p')
+        Tp = S2.tangent_space(p)
+        v = Tp((v_th0, v_ph0), basis=epolar_ON.at(p))
+        c = S2.integrated_autoparallel_curve(nab, (t, tmin, tmax), v,
+                                             chart=polar, name='c')
+        R3 = Manifold(3, 'R3', start_index=1)
+        cart = R3.chart('X Y Z')
+        euclid_embedding = S2.diff_map(R3,
+              {(polar, cart):[sin(th)*cos(ph),sin(th)*sin(ph),cos(th)]})
+        sol = c.solve(solution_key='sol-angle',
+            parameters_values={tmin:0,tmax:2,th0:pi/4,ph0:0.1,v_th0:1,v_ph0:8})
+        interp = c.interpolate(solution_key='sol-angle',
+                               interpolation_key='interp-angle')
+        graph3D_embedded_angle_curve = c.plot_integrated(interpolation_key='interp-angle',
+                                  mapping=euclid_embedding, thickness=3)
+        expr_ph = ph0+v_ph0/v_th0*(ln(tan((v_th0*t+th0)/2))-ln(tan(th0/2)))
+        c_loxo = S2.curve({polar: [th0+v_th0*t, expr_ph]}, (t,0,2), name='c_loxo')
+        graph3D_embedded_loxo = c_loxo.plot(mapping=euclid_embedding,
+              parameters={th0:pi/4+2*pi/180, ph0:0.1, v_th0:1, v_ph0:8},
+              thickness=3, color='blue')
+        graph3D_embedded_polar_coords = polar.plot(chart=cart,
+             mapping=euclid_embedding, number_values=15, color='yellow')
+        graph = graph3D_embedded_angle_curve + graph3D_embedded_loxo
+        graph += graph3D_embedded_polar_coords
+        sphinx_plot(graph)
+
+    """
+
+    def __init__(self, parent, affine_connection, curve_parameter,
+                 initial_tangent_vector, chart=None, name=None,
+                 latex_name=None, verbose=False):
+        r"""
+        Construct an autoparallel curve with respect to the given affine
+        connection with the given initial tangent vector.
+
+        TESTS::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, A, B] = var('t A B')
+            sage: nab = M.affine_connection('nabla', r'\nabla')
+            sage: nab[X.frame(),0,0,1],nab[X.frame(),2,1,2]=A*x1^2,B*x2*x3
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_autoparallel_curve(nab, (t, 0, 5), v,
+            ....:                                          name='c') ; c
+            Integrated autoparallel curve c in the 3-dimensional
+             differentiable manifold M
+            sage: TestSuite(c).run()
+
+        """
+
+        # setting the chart to gain access to the coordinate functions
+        if chart is None:
+            chart = parent.codomain().default_chart()
+
+        coordinate_functions = chart[:]
+        velocities = chart.symbolic_velocities()
+
+
+        dim = parent.codomain().dim()
+        i0 = parent.codomain().start_index()
+        equations_rhs = []
+
+        gamma = affine_connection.coef(frame=chart.frame())
+
+        for rho in range(dim):
+            rhs = 0
+            for mu in range(dim):
+                for nu in range(dim):
+                    vMUvNU = velocities[mu] * velocities[nu]
+                    gammaRHO_mu_nu = gamma[[rho+i0, mu+i0, nu+i0]].expr(chart=chart)
+                    # line above is the expression of the scalar
+                    # field 'gamma[[rho+i0, mu+i0, nu+i0]]' in terms
+                    # of 'chart' (here, in any point of the manifold,
+                    # the scalar field 'gamma[[rho+i0, mu+i0, nu+i0]]'
+                    # provides the coefficient [rho+i0, mu+i0, nu+i0]
+                    # of the affine connection with respect to frame
+                    # 'chart.frame()')
+                    rhs -= gammaRHO_mu_nu * vMUvNU
+                    # 'vMUvNU' and 'gammaRHO_mu_nu' only used for the
+                    # line above to be shorter
+            equations_rhs += [rhs.simplify_full()]
+
+        IntegratedCurve.__init__(self, parent, equations_rhs,
+                                 velocities, curve_parameter,
+                                 initial_tangent_vector, chart=chart,
+                                 name=name, latex_name=latex_name,
+                                 verbose=verbose)
+
+        self._affine_connection = affine_connection
+
+    def _repr_(self):
+        r"""
+        Return a string representation of ``self``.
+
+        TESTS::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, A, B] = var('t A B')
+            sage: nab = M.affine_connection('nabla', r'\nabla')
+            sage: nab[X.frame(),0,0,1],nab[X.frame(),2,1,2]=A*x1^2,B*x2*x3
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_autoparallel_curve(nab, (t,0,5), v); c
+            Integrated autoparallel curve in the 3-dimensional
+             differentiable manifold M
+            sage: c = M.integrated_autoparallel_curve(nab, (t, 0, 5), v,
+            ....:                                          name='c') ; c
+            Integrated autoparallel curve c in the 3-dimensional
+             differentiable manifold M
+
+        """
+
+        description = "Integrated autoparallel curve "
+        if self._name is not None:
+            description += self._name + " "
+        description += "in the {}".format(self._codomain)
+        return description
+
+    def __reduce__(self):
+        r"""
+        Reduction function for the pickle protocole.
+
+        TESTS::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, A, B] = var('t A B')
+            sage: nab = M.affine_connection('nabla', r'\nabla')
+            sage: nab[X.frame(),0,0,1],nab[X.frame(),2,1,2]=A*x1^2,B*x2*x3
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_autoparallel_curve(nab, (t, 0, 5), v,
+            ....:                                              name='c')
+            sage: c.__reduce__()
+            (<class 'sage.manifolds.differentiable.manifold_homset.IntegratedAutoparallelCurveSet_with_category.element_class'>,
+             (Set of Morphisms from Real interval (0, 5) to
+              3-dimensional differentiable manifold M in Category of
+              homsets of subobjects of sets and topological spaces which
+              actually are integrated autoparallel curves with respect to
+              a certain affine connection,
+              Affine connection nabla on the 3-dimensional
+              differentiable manifold M,
+              t,
+              Tangent vector at Point p on the 3-dimensional
+               differentiable manifold M,
+              Chart (M, (x1, x2, x3)),
+              'c',
+              'c'))
+
+        Test of pickling::
+
+            sage: loads(dumps(c))
+            Integrated autoparallel curve c in the 3-dimensional differentiable manifold M
+
+        """
+
+        return (type(self), (self.parent(), self._affine_connection,
+                self._curve_parameter, self._initial_tangent_vector,
+                self._chart, self._name, self._latex_name))
+
+    def system(self, verbose=False):
+        r"""
+        Provide a detailed description of the system defining the
+        autoparallel curve and returns the system defining it: chart,
+        equations and initial conditions.
+
+        INPUT:
+
+        - ``verbose`` -- (default: ``False``) prints a detailed
+          description of the curve
+
+        OUTPUT:
+
+        - list containing the
+
+          * the equations
+          * the initial conditions
+          * the chart
+
+        EXAMPLES:
+
+        System defining an autoparallel curve::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: [t, A, B] = var('t A B')
+            sage: nab = M.affine_connection('nabla', r'\nabla')
+            sage: nab[X.frame(),0,0,1],nab[X.frame(),2,1,2]=A*x1^2,B*x2*x3
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_autoparallel_curve(nab, (t, 0, 5), v)
+            sage: sys = c.system(verbose=True)
+            Autoparallel curve in the 3-dimensional differentiable
+             manifold M equipped with Affine connection nabla on the
+             3-dimensional differentiable manifold M, and integrated
+             over the Real interval (0, 5) as a solution to the
+             following equations, written with respect to
+             Chart (M, (x1, x2, x3)):
+            <BLANKLINE>
+            Initial point: Point p on the 3-dimensional differentiable
+             manifold M with coordinates [0, 0, 0] with respect to
+             Chart (M, (x1, x2, x3))
+            Initial tangent vector: Tangent vector at Point p on the
+             3-dimensional differentiable manifold M with
+             components [1, 0, 1] with respect to Chart (M, (x1, x2, x3))
+            <BLANKLINE>
+            d(x1)/dt = Dx1
+            d(x2)/dt = Dx2
+            d(x3)/dt = Dx3
+            d(Dx1)/dt = -A*Dx1*Dx2*x1^2
+            d(Dx2)/dt = 0
+            d(Dx3)/dt = -B*Dx2*Dx3*x2*x3
+            <BLANKLINE>
+            sage: sys_bis = c.system()
+            sage: sys_bis == sys
+            True
+
+        """
+
+        v0 = self._initial_tangent_vector
+        chart = self._chart
+
+        if verbose:
+            initial_tgt_space = v0.parent()
+            initial_pt = initial_tgt_space.base_point() # retrieves
+            # the initial point as the base point of the tangent space
+            # to which initial tangent vector belongs
+            initial_pt_coords = list(initial_pt.coordinates(chart))
+            # previous line converts to list since would otherwise be a
+            # tuple ; will raise error if coordinates in chart are not
+            # known
+
+            initial_coord_basis = chart.frame().at(initial_pt)
+            initial_tgt_vec_comps = v0[initial_coord_basis,:] # will
+            # raise error if components in coordinate basis are not
+            # known
+
+            description = "Autoparallel curve "
+            if self._name is not None:
+                description += self._name + " "
+            description += "in the {} ".format(self.codomain())
+            description += "equipped with "
+            description += "{}, ".format(self._affine_connection)
+            description += "and integrated over the "
+            description += "{} ".format(self.domain())
+            description += "as a solution to the following equations, "
+            description += "written with respect to "
+            description += "{}:\n\n".format(chart)
+
+            description += "Initial point: {} ".format(initial_pt)
+            description += "with coordinates "
+            description += "{} ".format(initial_pt_coords)
+            description += "with respect to {}\n".format(chart)
+
+            description += "Initial tangent vector: {} ".format(v0)
+            description += "with components "
+            description +="{}".format(initial_tgt_vec_comps)
+            description += " with respect to {}\n\n".format(chart)
+
+            for coord_func,velocity in zip(chart[:],self._velocities):
+                description += "d({})/d{} = {}\n".format(coord_func,
+                                                  self._curve_parameter,
+                                                  velocity)
+
+            for velocity,eqn in zip(self._velocities,self._equations_rhs):
+                description += "d({})/d{} = {}\n".format(velocity,
+                                                  self._curve_parameter,
+                                                  eqn)
+
+            print(description)
+
+        return [self._equations_rhs, v0, chart]
+
+class IntegratedGeodesic(IntegratedAutoparallelCurve):
+    r"""
+    Geodesic on the manifold with respect to a given metric.
+
+    INPUT:
+
+    - ``parent`` --
+      :class:`~sage.manifolds.differentiable.manifold_homset.IntegratedGeodesicSet`
+      the set of curves `\mathrm{Hom_{geodesic}}(I, M)` to which the
+      curve belongs
+    - ``metric`` --
+      :class:`~sage.manifolds.differentiable.metric.PseudoRiemannianMetric`
+      metric with respect to which the curve is a geodesic
+    - ``curve_parameter`` -- symbolic expression to be used as the
+      parameter of the curve (the equations defining an instance of
+      IntegratedGeodesic are such that ``t`` will actually be an affine
+      parameter of the curve);
+    - ``initial_tangent_vector`` --
+      :class:`~sage.manifolds.differentiable.tangent_vector.TangentVector`
+      initial tangent vector of the curve
+    - ``chart`` -- (default: ``None``) chart on the manifold in terms of
+      which the equations are expressed; if ``None`` the default chart
+      of the manifold is assumed
+    - ``name`` -- (default: ``None``) string; symbol given to the curve
+    - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote
+      the curve; if none is provided, ``name`` will be used
+
+    EXAMPLES:
+
+    Geodesics of the unit 2-sphere `\mathbb{S}^{2}`.
+    Start with declaring the standard polar coordinates
+    `(\theta, \phi)` on `\mathbb{S}^{2}` and the
+    corresponding coordinate frame `(e_{\theta}, e_{\phi})`::
+
+        sage: S2 = Manifold(2, 'S^2', start_index=1)
+        sage: polar.<th,ph>=S2.chart('th ph')
+        sage: epolar = polar.frame()
+
+    Set the Euclidean metric tensor `g` induced on `\mathbb{S}^{2}`::
+
+        sage: g = S2.metric('g')
+        sage: g[1,1], g[2,2] = 1, (sin(th))^2
+
+    Set generic initial conditions for the geodesics to compute::
+
+        sage: [th0, ph0, v_th0, v_ph0] = var('th0 ph0 v_th0 v_ph0')
+        sage: p = S2.point((th0, ph0), name='p')
+        sage: Tp = S2.tangent_space(p)
+        sage: v = Tp((v_th0, v_ph0), basis=epolar.at(p))
+
+    Declare the corresponding integrated geodesic and display the
+    differential system it satisfies::
+
+        sage: [t, tmin, tmax] = var('t tmin tmax')
+        sage: c = S2.integrated_geodesic(g, (t, tmin, tmax), v,
+        ....:                            chart=polar, name='c')
+        sage: sys = c.system(verbose=True)
+        Geodesic c in the 2-dimensional differentiable manifold S^2
+         equipped with Riemannian metric g on the 2-dimensional
+         differentiable manifold S^2, and integrated over the Real
+         interval (tmin, tmax) as a solution to the following geodesic
+         equations, written with respect to Chart (S^2, (th, ph)):
+        <BLANKLINE>
+        Initial point: Point p on the 2-dimensional differentiable
+        manifold S^2 with coordinates [th0, ph0] with respect to
+        Chart (S^2, (th, ph))
+        Initial tangent vector: Tangent vector at Point p on the
+        2-dimensional differentiable manifold S^2 with
+        components [v_th0, v_ph0] with respect to Chart (S^2, (th, ph))
+        <BLANKLINE>
+        d(th)/dt = Dth
+        d(ph)/dt = Dph
+        d(Dth)/dt = Dph^2*cos(th)*sin(th)
+        d(Dph)/dt = -2*Dph*Dth*cos(th)/sin(th)
+        <BLANKLINE>
+
+    Set a dictionnary providing the parameter range and the initial
+    conditions for various geodesics::
+
+        sage: dict_params={'equat':{tmin:0,tmax:3,th0:pi/2,ph0:0.1,v_th0:0,v_ph0:1},
+        ....:   'longi':{tmin:0,tmax:3,th0:0.1,ph0:0.1,v_th0:1,v_ph0:0},
+        ....:   'angle':{tmin:0,tmax:3,th0:pi/4,ph0:0.1,v_th0:1,v_ph0:1}}
+
+    Use `\mathbb{R}^{3}` as the codomain of the standard map
+    embedding `(\mathbb{S}^{2}, (\theta, \phi))` in the
+    3-dimensional Euclidean space::
+
+        sage: R3 = Manifold(3, 'R3', start_index=1)
+        sage: cart.<X,Y,Z> = R3.chart()
+        sage: euclid_embedding = S2.diff_map(R3,
+        ....:  {(polar, cart):[sin(th)*cos(ph),sin(th)*sin(ph),cos(th)]})
+
+    Solve, interpolate and prepare the plot for the solutions
+    corresponding to the three initial conditions previously set::
+
+        sage: graph3D_embedded_geods = Graphics()
+        sage: for key in dict_params:
+        ....:     sol = c.solve(solution_key='sol-'+key,
+        ....:                        parameters_values=dict_params[key])
+        ....:     interp = c.interpolate(solution_key='sol-'+key,
+        ....:                           interpolation_key='interp-'+key)
+        ....:     graph3D_embedded_geods += c.plot_integrated(interpolation_key='interp-'+key,
+        ....:                      mapping=euclid_embedding, thickness=5,
+        ....:                      display_tangent=True, scale=0.3,
+        ....:                      width_tangent=0.5)
+
+    Plot the resulting geodesics on the grid of polar coordinates lines
+    on `\mathbb{S}^{2}` and check that these are great circles::
+
+        sage: graph3D_embedded_polar_coords = polar.plot(chart=cart,
+        ....:                          mapping=euclid_embedding,
+        ....:                          number_values=15, color='yellow')
+        sage: graph3D_embedded_geods + graph3D_embedded_polar_coords
+        Graphics3d Object
+
+    .. PLOT::
+
+        S2 = Manifold(2, 'S^2', start_index=1)
+        polar = S2.chart('th ph'); th, ph = polar[:]
+        epolar = polar.frame()
+        g = S2.metric('g')
+        g[1,1], g[2,2] = 1, (sin(th))**2
+        t,tmin,tmax,th0,ph0,v_th0,v_ph0 = var('t tmin tmax th0 ph0 v_th0 v_ph0')
+        p = S2.point((th0, ph0), name='p')
+        Tp = S2.tangent_space(p)
+        v = Tp((v_th0, v_ph0), basis=epolar.at(p))
+        c = S2.integrated_geodesic(g, (t, tmin, tmax), v, chart=polar,
+                                                               name='c')
+        dict_params={'equat':{tmin:0,tmax:3,th0:pi/2,ph0:0.1,v_th0:0,v_ph0:1},
+                     'longi':{tmin:0,tmax:3,th0:0.1,ph0:0.1,v_th0:1,v_ph0:0},
+                     'angle':{tmin:0,tmax:3,th0:pi/4,ph0:0.1,v_th0:1,v_ph0:1}}
+        R3 = Manifold(3, 'R3', start_index=1)
+        cart = R3.chart('X Y Z')
+        euclid_embedding = S2.diff_map(R3,
+              {(polar, cart): [sin(th)*cos(ph),sin(th)*sin(ph),cos(th)]})
+        graph3D_embedded_geods = Graphics()
+        for key in dict_params:
+            sol = c.solve(solution_key='sol-'+key,
+                          parameters_values=dict_params[key])
+            interp = c.interpolate(solution_key='sol-'+key,
+                                   interpolation_key='interp-'+key)
+            graph3D_embedded_geods += c.plot_integrated(interpolation_key='interp-'+key,
+                                  mapping=euclid_embedding, thickness=5,
+                                  display_tangent=True, scale=0.3,
+                                  width_tangent=0.5)
+        graph3D_embedded_polar_coords = polar.plot(chart=cart,
+                                       mapping=euclid_embedding,
+                                       number_values=15, color='yellow')
+        graph = graph3D_embedded_geods + graph3D_embedded_polar_coords
+        sphinx_plot(graph)
+
+    """
+
+    def __init__(self, parent, metric, curve_parameter,
+                 initial_tangent_vector, chart=None, name=None,
+                 latex_name=None, verbose=False):
+
+        r"""
+        Construct a geodesic curve with respect to the given metric with the
+        given initial tangent vector.
+
+        TESTS::
+
+            sage: S2 = Manifold(2, 'S^2')
+            sage: X.<theta,phi> = S2.chart()
+            sage: [t, A] = var('t A')
+            sage: g = S2.metric('g')
+            sage: g[0,0] = A
+            sage: g[1,0] = 0
+            sage: g[1,1] = A*sin(theta)^2
+            sage: p = S2.point((pi/2,0), name='p')
+            sage: Tp = S2.tangent_space(p)
+            sage: v = Tp((1/sqrt(2),1/sqrt(2)))
+            sage: c = S2.integrated_geodesic(g, (t,0,pi), v, name='c'); c
+            Integrated geodesic c in the 2-dimensional differentiable
+             manifold S^2
+            sage: TestSuite(c).run()
+
+        """
+
+        affine_connection = metric.connection()
+
+        IntegratedAutoparallelCurve.__init__(self, parent,
+                                             affine_connection, curve_parameter,
+                                             initial_tangent_vector, chart=chart,
+                                             name=name, latex_name=latex_name,
+                                             verbose=verbose)
+
+        self._metric = metric
+
+    def _repr_(self):
+        r"""
+        Return a string representation of ``self``.
+
+        TESTS::
+
+            sage: S2 = Manifold(2, 'S^2')
+            sage: X.<theta,phi> = S2.chart()
+            sage: [t, A] = var('t A')
+            sage: g = S2.metric('g')
+            sage: g[0,0] = A
+            sage: g[1,0] = 0
+            sage: g[1,1] = A*sin(theta)^2
+            sage: p = S2.point((pi/2,0), name='p')
+            sage: Tp = S2.tangent_space(p)
+            sage: v = Tp((1/sqrt(2),1/sqrt(2)))
+            sage: c = S2.integrated_geodesic(g, (t, 0, pi), v) ; c
+            Integrated geodesic in the 2-dimensional differentiable
+             manifold S^2
+            sage: c = S2.integrated_geodesic(g, (t,0,pi), v, name='c'); c
+            Integrated geodesic c in the 2-dimensional differentiable
+             manifold S^2
+
+        """
+
+        description = "Integrated geodesic "
+        if self._name is not None:
+            description += self._name + " "
+        description += "in the {}".format(self._codomain)
+        return description
+
+    def __reduce__(self):
+        r"""
+        Reduction function for the pickle protocole.
+
+        TESTS::
+
+            sage: S2 = Manifold(2, 'S^2')
+            sage: X.<theta,phi> = S2.chart()
+            sage: [t, A] = var('t A')
+            sage: g = S2.metric('g')
+            sage: g[0,0] = A
+            sage: g[1,0] = 0
+            sage: g[1,1] = A*sin(theta)^2
+            sage: p = S2.point((pi/2,0), name='p')
+            sage: Tp = S2.tangent_space(p)
+            sage: v = Tp((1/sqrt(2),1/sqrt(2)))
+            sage: c = S2.integrated_geodesic(g, (t, 0, pi), v, name='c')
+            sage: c.__reduce__()
+            (<...IntegratedGeodesicSet_with_category.element_class'>,
+             (Set of Morphisms from Real interval (0, pi) to
+              2-dimensional differentiable manifold S^2 in Category of
+              homsets of subobjects of sets and topological spaces which
+              actually are integrated geodesics with respect to a certain
+              metric,
+              Riemannian metric g on the 2-dimensional differentiable
+              manifold S^2,
+              t,
+              Tangent vector at Point p on the 2-dimensional
+               differentiable manifold S^2,
+              Chart (S^2, (theta, phi)),
+              'c',
+              'c'))
+
+        Test of pickling::
+
+            sage: loads(dumps(c))
+            Integrated geodesic c in the 2-dimensional differentiable manifold S^2
+
+        """
+
+        return (type(self), (self.parent(), self._metric,
+                self._curve_parameter, self._initial_tangent_vector,
+                self._chart, self._name, self._latex_name))
+
+    def system(self, verbose=False):
+        r"""
+        Return the system defining the geodesic: chart, equations and
+        initial conditions.
+
+        INPUT:
+
+        - ``verbose`` -- (default: ``False``) prints a detailed
+          description of the curve
+
+        OUTPUT:
+
+        - list containing
+
+          * the equations
+          * the initial equations
+          * the chart
+
+        EXAMPLES:
+
+        System defining a geodesic::
+
+            sage: S2 = Manifold(2, 'S^2')
+            sage: X.<theta,phi> = S2.chart()
+            sage: [t, A] = var('t A')
+            sage: g = S2.metric('g')
+            sage: g[0,0] = A
+            sage: g[1,0] = 0
+            sage: g[1,1] = A*sin(theta)^2
+            sage: p = S2.point((pi/2,0), name='p')
+            sage: Tp = S2.tangent_space(p)
+            sage: v = Tp((1/sqrt(2),1/sqrt(2)))
+            sage: c = S2.integrated_geodesic(g, (t, 0, pi), v, name='c')
+            sage: sys = c.system(verbose=True)
+            Geodesic c in the 2-dimensional differentiable manifold S^2
+             equipped with Riemannian metric g on the 2-dimensional
+             differentiable manifold S^2, and integrated over the Real
+             interval (0, pi) as a solution to the following geodesic
+             equations, written with respect to Chart (S^2, (theta, phi)):
+            <BLANKLINE>
+            Initial point: Point p on the 2-dimensional differentiable
+             manifold S^2 with coordinates [1/2*pi, 0] with respect to
+             Chart (S^2, (theta, phi))
+            Initial tangent vector: Tangent vector at Point p on the
+             2-dimensional differentiable manifold S^2 with
+             components [1/2*sqrt(2), 1/2*sqrt(2)] with respect to
+             Chart (S^2, (theta, phi))
+            <BLANKLINE>
+            d(theta)/dt = Dtheta
+            d(phi)/dt = Dphi
+            d(Dtheta)/dt = Dphi^2*cos(theta)*sin(theta)
+            d(Dphi)/dt = -2*Dphi*Dtheta*cos(theta)/sin(theta)
+            <BLANKLINE>
+            sage: sys_bis = c.system()
+            sage: sys_bis == sys
+            True
+
+        """
+
+        v0 = self._initial_tangent_vector
+        chart = self._chart
+
+        if verbose:
+            initial_tgt_space = v0.parent()
+            initial_pt = initial_tgt_space.base_point()#retrieves
+            # the initial point as the base point of the tangent space
+            # to which initial tangent vector belongs
+            initial_pt_coords = list(initial_pt.coordinates(chart))
+            # previous line converts to list since would otherwise be a
+            # tuple ; will raise error if coordinates in chart are not
+            # known
+
+            initial_coord_basis = chart.frame().at(initial_pt)
+            initial_tgt_vec_comps = v0[initial_coord_basis,:] # will
+            # raise error if components in coordinate basis are not
+            # known
+
+            description = "Geodesic "
+            if self._name is not None:
+                description += self._name + " "
+            description += "in the {} ".format(self.codomain())
+            description += "equipped with "
+            description += "{}, ".format(self._metric)
+            description += "and integrated over the "
+            description += "{} ".format(self.domain())
+            description += "as a solution to the following "
+            description += "geodesic equations, written with respect to "
+            description += "{}:\n\n".format(chart)
+
+            description += "Initial point: {} ".format(initial_pt)
+            description += "with coordinates "
+            description += "{} ".format(initial_pt_coords)
+            description += "with respect to {}\n".format(chart)
+
+            description += "Initial tangent vector: {} ".format(v0)
+            description += "with components "
+            description +="{}".format(initial_tgt_vec_comps)
+            description += " with respect to {}\n\n".format(chart)
+
+            for coord_func,velocity in zip(chart[:],self._velocities):
+                description += "d({})/d{} = {}\n".format(coord_func,
+                                                  self._curve_parameter,
+                                                  velocity)
+
+            for velocity,eqn in zip(self._velocities,self._equations_rhs):
+                description += "d({})/d{} = {}\n".format(velocity,
+                                                  self._curve_parameter,
+                                                  eqn)
+
+            print(description)
+
+        return [self._equations_rhs, v0, chart]
+
diff --git a/src/sage/manifolds/differentiable/levi_civita_connection.py b/src/sage/manifolds/differentiable/levi_civita_connection.py
index 0e6727b1b70..332ed94dc51 100644
--- a/src/sage/manifolds/differentiable/levi_civita_connection.py
+++ b/src/sage/manifolds/differentiable/levi_civita_connection.py
@@ -38,7 +38,7 @@ class LeviCivitaConnection(AffineConnection):
     Levi-Civita connection on a pseudo-Riemannian manifold.
 
     Let `M` be a differentiable manifold of class `C^\infty` (smooth manifold)
-    over `\RR` endowed with with a pseudo-Riemannian metric `g`.
+    over `\RR` endowed with a pseudo-Riemannian metric `g`.
     Let `C^\infty(M)` be the algebra of smooth functions
     `M\rightarrow \RR` (cf.
     :class:`~sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra`)
diff --git a/src/sage/manifolds/differentiable/manifold.py b/src/sage/manifolds/differentiable/manifold.py
index 609e28006c7..e470a39e42f 100644
--- a/src/sage/manifolds/differentiable/manifold.py
+++ b/src/sage/manifolds/differentiable/manifold.py
@@ -333,7 +333,7 @@
     sage: i = M((I,), chart=Z, name='i'); i
     Point i on the 1-dimensional complex manifold C*
 
-Its coordinates w.r.t. the charts ``Z`` and ``W`` are::
+Its coordinates with respect to the charts ``Z`` and ``W`` are::
 
     sage: Z(i)
     (I,)
@@ -803,8 +803,8 @@ def open_subset(self, name, latex_name=None, coord_def={}):
             sage: p in U
             True
 
-        Checking whether various points, defined by their coordinates w.r.t.
-        chart ``X``,  are in ``U``::
+        Checking whether various points, defined by their coordinates
+        with respect to chart ``X``,  are in ``U``::
 
             sage: M((0,1/2)) in U
             True
@@ -2570,6 +2570,371 @@ def curve(self, coord_expression, param, chart=None,
                 coord_expression = {chart: (coord_expression,)}
         return curve_set(coord_expression, name=name, latex_name=latex_name)
 
+    def integrated_curve(self, equations_rhs, velocities, curve_param,
+                         initial_tangent_vector, chart=None, name=None,
+                         latex_name=None, verbose=False):
+        r"""
+        Construct a curve defined by a system of second order
+        differential equations in the coordinate functions.
+
+        .. SEEALSO::
+
+            :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedCurve`
+            for details.
+
+        INPUT:
+
+        - ``equations_rhs`` -- list of the right-hand sides of the
+          equations on the velocities only
+        - ``velocities`` -- list of the symbolic expressions used in
+          ``equations_rhs`` to denote the velocities
+        - ``curve_param`` -- a tuple of the type ``(t, t_min, t_max)``,
+          where
+
+          * ``t`` is the symbolic variable used in ``equations_rhs`` to
+            denote the parameter of the curve;
+          * ``t_min`` is its minimal (finite) value;
+          * ``t_max`` its maximal (finite) value.
+
+        - ``initial_tangent_vector`` --
+          :class:`~sage.manifolds.differentiable.tangent_vector.TangentVector`;
+          initial tangent vector of the curve
+        - ``chart`` -- (default: ``None``) chart on the manifold in
+          which the equations are given; if ``None`` the default chart
+          of the manifold is assumed
+        - ``name`` -- (default: ``None``) string; symbol given to the curve
+        - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote
+          the curve; if none is provided, ``name`` will be used
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedCurve`
+
+        EXAMPLES:
+
+        Trajectory of a particle of unit mass and unit charge in a
+        unit, uniform, stationary magnetic field::
+
+            sage: M = Manifold(3, 'M')
+            sage: X.<x1,x2,x3> = M.chart()
+            sage: t = var('t')
+            sage: D = X.symbolic_velocities()
+            sage: eqns = [D[1], -D[0], SR(0)]
+            sage: p = M.point((0,0,0), name='p')
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,0,1))
+            sage: c = M.integrated_curve(eqns, D, (t,0,6), v, name='c'); c
+            Integrated curve c in the 3-dimensional differentiable
+             manifold M
+            sage: sys = c.system(verbose=True)
+            Curve c in the 3-dimensional differentiable manifold M
+             integrated over the Real interval (0, 6) as a solution to
+             the following system, written with respect to
+             Chart (M, (x1, x2, x3)):
+            <BLANKLINE>
+            Initial point: Point p on the 3-dimensional differentiable
+             manifold M with coordinates [0, 0, 0] with respect to
+             Chart (M, (x1, x2, x3))
+            Initial tangent vector: Tangent vector at Point p on the
+             3-dimensional differentiable manifold M with
+             components [1, 0, 1] with respect to Chart (M, (x1, x2, x3))
+            <BLANKLINE>
+            d(x1)/dt = Dx1
+            d(x2)/dt = Dx2
+            d(x3)/dt = Dx3
+            d(Dx1)/dt = Dx2
+            d(Dx2)/dt = -Dx1
+            d(Dx3)/dt = 0
+            <BLANKLINE>
+            sage: sol = c.solve()
+            sage: interp = c.interpolate()
+            sage: p = c(1.3, verbose=True)
+            Evaluating point coordinates from the interpolation
+             associated with the key 'cubic spline-interp-rk4_maxima'
+             by default...
+            sage: p
+            Point on the 3-dimensional differentiable manifold M
+            sage: p.coordinates()     # abs tol 1e-12
+            (0.9635581155730744, -0.7325010457963622, 1.3)
+            sage: tgt_vec = c.tangent_vector_eval_at(3.7, verbose=True)
+            Evaluating tangent vector components from the interpolation
+             associated with the key 'cubic spline-interp-rk4_maxima'
+             by default...
+            sage: tgt_vec[:]     # abs tol 1e-12
+            [-0.8481008455360024, 0.5298346120470748, 1.0000000000000007]
+
+        """
+
+        from sage.manifolds.differentiable.real_line import RealLine
+        from sage.manifolds.differentiable.manifold_homset import IntegratedCurveSet
+
+        if len(curve_param) != 3:
+            raise ValueError("the argument 'curve_param' must be of the form " +
+                             "(t, t_min, t_max)")
+        t = curve_param[0]
+        t_min = curve_param[1]
+        t_max = curve_param[2]
+        real_field = RealLine(names=(repr(t),))
+        interval = real_field.open_interval(t_min, t_max)
+        integrated_curve_set = IntegratedCurveSet(interval, self) # not
+        # possible to use Hom(interval, self)
+        return integrated_curve_set(equations_rhs, velocities, t,
+                                    initial_tangent_vector, chart=chart,
+                                    name=name, latex_name=latex_name,
+                                    verbose=verbose)
+
+    def integrated_autoparallel_curve(self, affine_connection,
+                        curve_param, initial_tangent_vector, chart=None,
+                        name=None, latex_name=None, verbose=False):
+        r"""
+        Construct an autoparallel curve on the manifold with respect to
+        a given affine connection.
+
+        .. SEEALSO::
+
+            :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedAutoparallelCurve`
+            for details.
+
+        INPUT:
+
+        - ``affine_connection`` --
+          :class:`~sage.manifolds.differentiable.affine_connection.AffineConnection`;
+          affine connection with respect to which the curve is autoparallel
+        - ``curve_param`` -- a tuple of the type ``(t, t_min, t_max)``,
+          where
+
+          * ``t`` is the symbolic variable to be used as the parameter
+            of the curve (the equations defining an instance of
+            :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedAutoparallelCurve`
+            are such that ``t`` will actually be an affine parameter
+            of the curve);
+          * ``t_min`` is its minimal (finite) value;
+          * ``t_max`` its maximal (finite) value.
+
+        - ``initial_tangent_vector`` --
+          :class:`~sage.manifolds.differentiable.tangent_vector.TangentVector`;
+          initial tangent vector of the curve
+        - ``chart`` -- (default: ``None``) chart on the manifold in
+          which the equations are given ; if ``None`` the default chart
+          of the manifold is assumed
+        - ``name`` -- (default: ``None``) string; symbol given to the curve
+        - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote
+          the curve; if none is provided, ``name`` will be used
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedAutoparallelCurve`
+
+        EXAMPLES:
+
+        Autoparallel curves associated with the Mercator projection of
+        the 2-sphere `\mathbb{S}^{2}`::
+
+            sage: S2 = Manifold(2, 'S^2', start_index=1)
+            sage: polar.<th,ph> = S2.chart('th ph')
+            sage: epolar = polar.frame()
+            sage: ch_basis = S2.automorphism_field()
+            sage: ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
+            sage: epolar_ON=S2.default_frame().new_frame(ch_basis,'epolar_ON')
+
+        Set the affine connection associated with Mercator projection;
+        it is metric compatible but it has non-vanishing torsion::
+
+            sage: nab = S2.affine_connection('nab')
+            sage: nab.set_coef(epolar_ON)[:]
+            [[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
+            sage: g = S2.metric('g')
+            sage: g[1,1], g[2,2] = 1, (sin(th))^2
+            sage: nab(g)[:]
+            [[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
+            sage: nab.torsion()[:]
+            [[[0, 0], [0, 0]], [[0, cos(th)/sin(th)], [-cos(th)/sin(th), 0]]]
+
+        Declare an integrated autoparallel curve with respect to this
+        connection::
+
+            sage: p = S2.point((pi/4, 0), name='p')
+            sage: Tp = S2.tangent_space(p)
+            sage: v = Tp((1,1), basis=epolar_ON.at(p))
+            sage: t = var('t')
+            sage: c = S2.integrated_autoparallel_curve(nab, (t, 0, 6),
+            ....:                              v, chart=polar, name='c')
+            sage: sys = c.system(verbose=True)
+            Autoparallel curve c in the 2-dimensional differentiable
+             manifold S^2 equipped with Affine connection nab on the
+             2-dimensional differentiable manifold S^2, and integrated
+             over the Real interval (0, 6) as a solution to the
+             following equations, written with respect to
+             Chart (S^2, (th, ph)):
+            <BLANKLINE>
+            Initial point: Point p on the 2-dimensional differentiable
+             manifold S^2 with coordinates [1/4*pi, 0] with respect to
+             Chart (S^2, (th, ph))
+            Initial tangent vector: Tangent vector at Point p on the
+             2-dimensional differentiable manifold S^2 with
+             components [1, sqrt(2)] with respect to
+             Chart (S^2, (th, ph))
+            <BLANKLINE>
+            d(th)/dt = Dth
+            d(ph)/dt = Dph
+            d(Dth)/dt = 0
+            d(Dph)/dt = -Dph*Dth*cos(th)/sin(th)
+            <BLANKLINE>
+            sage: sol = c.solve()
+            sage: interp = c.interpolate()
+            sage: p = c(1.3, verbose=True)
+            Evaluating point coordinates from the interpolation
+             associated with the key 'cubic spline-interp-rk4_maxima'
+             by default...
+            sage: p
+            Point on the 2-dimensional differentiable manifold S^2
+            sage: p.coordinates()     # abs tol 1e-12
+            (2.085398163397449, 1.4203172015958863)
+            sage: tgt_vec = c.tangent_vector_eval_at(3.7, verbose=True)
+            Evaluating tangent vector components from the interpolation
+             associated with the key 'cubic spline-interp-rk4_maxima'
+             by default...
+            sage: tgt_vec[:]     # abs tol 1e-12
+            [0.9999999999999732, -1.016513736236512]
+
+        """
+
+        from sage.manifolds.differentiable.real_line import RealLine
+        from sage.manifolds.differentiable.manifold_homset import IntegratedAutoparallelCurveSet
+
+        if len(curve_param) != 3:
+            raise ValueError("the argument 'curve_param' must be " +
+                             "of the form (t, t_min, t_max)")
+        t = curve_param[0]
+        t_min = curve_param[1]
+        t_max = curve_param[2]
+        real_field = RealLine(names=(repr(t),))
+        interval = real_field.open_interval(t_min, t_max)
+        autoparallel_curve_set = IntegratedAutoparallelCurveSet(interval,
+                                                                self)
+        # not possible to use Hom(interval, self)
+        return autoparallel_curve_set(affine_connection, t,
+                                      initial_tangent_vector,
+                                      chart=chart, name=name,
+                                      latex_name=latex_name,
+                                      verbose=verbose)
+
+    def integrated_geodesic(self, metric, curve_param,
+                            initial_tangent_vector, chart=None,
+                            name=None, latex_name=None, verbose=False):
+        r"""
+        Construct a geodesic on the manifold with respect to a given metric.
+
+        .. SEEALSO::
+
+            :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedGeodesic`
+            for details.
+
+        INPUT:
+
+        - ``metric`` --
+          :class:`~sage.manifolds.differentiable.metric.PseudoRiemannianMetric`
+          metric with respect to which the curve is a geodesic
+        - ``curve_param`` -- a tuple of the type ``(t, t_min, t_max)``,
+          where
+
+          * ``t`` is the symbolic variable to be used as the parameter
+            of the curve (the equations defining an instance of
+            :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedGeodesic`
+            are such that ``t`` will actually be an affine parameter
+            of the curve);
+          * ``t_min`` is its minimal (finite) value;
+          * ``t_max`` its maximal (finite) value.
+
+        - ``initial_tangent_vector`` --
+          :class:`~sage.manifolds.differentiable.tangent_vector.TangentVector`;
+          initial tangent vector of the curve
+        - ``chart`` -- (default: ``None``) chart on the manifold in
+          which the equations are given; if ``None`` the default chart
+          of the manifold is assumed
+        - ``name`` -- (default: ``None``) string; symbol given to the curve
+        - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote
+          the curve; if none is provided, ``name`` will be used
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedGeodesic`
+
+        EXAMPLES:
+
+        Geodesics of the unit 2-sphere `\mathbb{S}^{2}`::
+
+            sage: S2 = Manifold(2, 'S^2', start_index=1)
+            sage: polar.<th,ph> = S2.chart('th ph')
+            sage: epolar = polar.frame()
+
+        Set the standard metric tensor `g` on `\mathbb{S}^{2}`::
+
+            sage: g = S2.metric('g')
+            sage: g[1,1], g[2,2] = 1, (sin(th))^2
+
+        Declare an integrated geodesic with respect to this metric::
+
+            sage: p = S2.point((pi/4, 0), name='p')
+            sage: Tp = S2.tangent_space(p)
+            sage: v = Tp((1, 1), basis=epolar.at(p))
+            sage: t = var('t')
+            sage: c = S2.integrated_geodesic(g, (t, 0, 6), v,
+            ....:                                 chart=polar, name='c')
+            sage: sys = c.system(verbose=True)
+            Geodesic c in the 2-dimensional differentiable manifold S^2
+             equipped with Riemannian metric g on the 2-dimensional
+             differentiable manifold S^2, and integrated over the Real
+             interval (0, 6) as a solution to the following geodesic
+             equations, written with respect to Chart (S^2, (th, ph)):
+            <BLANKLINE>
+            Initial point: Point p on the 2-dimensional differentiable
+            manifold S^2 with coordinates [1/4*pi, 0] with respect to
+            Chart (S^2, (th, ph))
+            Initial tangent vector: Tangent vector at Point p on the
+            2-dimensional differentiable manifold S^2 with
+            components [1, 1] with respect to Chart (S^2, (th, ph))
+            <BLANKLINE>
+            d(th)/dt = Dth
+            d(ph)/dt = Dph
+            d(Dth)/dt = Dph^2*cos(th)*sin(th)
+            d(Dph)/dt = -2*Dph*Dth*cos(th)/sin(th)
+            <BLANKLINE>
+            sage: sol = c.solve()
+            sage: interp = c.interpolate()
+            sage: p = c(1.3, verbose=True)
+            Evaluating point coordinates from the interpolation
+             associated with the key 'cubic spline-interp-rk4_maxima'
+             by default...
+            sage: p
+            Point on the 2-dimensional differentiable manifold S^2
+            sage: p.coordinates()     # abs tol 1e-12
+            (2.2047444794514663, 0.7986609561213334)
+            sage: tgt_vec = c.tangent_vector_eval_at(3.7, verbose=True)
+            Evaluating tangent vector components from the interpolation
+             associated with the key 'cubic spline-interp-rk4_maxima'
+             by default...
+            sage: tgt_vec[:]     # abs tol 1e-12
+            [-1.090742147346732, 0.620568327518154]
+
+        """
+        from sage.manifolds.differentiable.real_line import RealLine
+        from sage.manifolds.differentiable.manifold_homset import IntegratedGeodesicSet
+
+        if len(curve_param) != 3:
+            raise ValueError("the argument 'curve_param' must be of " +
+                             "the form (t, t_min, t_max)")
+        t = curve_param[0]
+        t_min = curve_param[1]
+        t_max = curve_param[2]
+        real_field = RealLine(names=(repr(t),))
+        interval = real_field.open_interval(t_min, t_max)
+        integrated_geodesic_set = IntegratedGeodesicSet(interval, self)
+        return integrated_geodesic_set(metric, t, initial_tangent_vector,
+                                       chart=chart, name=name,
+                                       latex_name=latex_name,
+                                       verbose=verbose)
+
     def affine_connection(self, name, latex_name=None):
         r"""
         Define an affine connection on the manifold.
diff --git a/src/sage/manifolds/differentiable/manifold_homset.py b/src/sage/manifolds/differentiable/manifold_homset.py
index ec9506a5c01..4a85bb8e9a4 100644
--- a/src/sage/manifolds/differentiable/manifold_homset.py
+++ b/src/sage/manifolds/differentiable/manifold_homset.py
@@ -1,19 +1,32 @@
 r"""
 Sets of Morphisms between Differentiable Manifolds
 
-The class :class:`DifferentiableManifoldHomset` implements sets of morphisms between
-two differentiable manifolds over the same topological field `K` (in most
-applications, `K = \RR` or `K = \CC`), a morphism being a *differentiable map*
-for the category of differentiable manifolds.
+The class :class:`DifferentiableManifoldHomset` implements sets of morphisms
+between two differentiable manifolds over the same topological field `K`
+(in most applications, `K = \RR` or `K = \CC`), a morphism being a
+*differentiable map* for the category of differentiable manifolds.
 
-The subclass :class:`DifferentiableCurveSet` is devoted to the specific case of
-differential curves, i.e. morphisms whose domain is an open interval of
+The subclass :class:`DifferentiableCurveSet` is devoted to the specific case
+of differential curves, i.e. morphisms whose domain is an open interval of
 `\RR`.
 
+The subclass :class:`IntegratedCurveSet` is devoted to differentiable
+curves that are defined as a solution to a system of second order
+differential equations.
+
+The subclass :class:`IntegratedAutoparallelCurveSet` is devoted to
+differentiable curves that are defined as autoparallel curves with respect to
+a certain affine connection.
+
+The subclass :class:`IntegratedGeodesicSet` is devoted to differentiable
+curves that are defined as geodesics with respect to to a certain metric.
+
 AUTHORS:
 
 - Eric Gourgoulhon (2015): initial version
 - Travis Scrimshaw (2016): review tweaks
+- Karim Van Aelst (2017): sets of integrated curves
+
 
 REFERENCES:
 
@@ -33,6 +46,10 @@
 from sage.manifolds.manifold_homset import TopologicalManifoldHomset
 from sage.manifolds.differentiable.diff_map import DiffMap
 from sage.manifolds.differentiable.curve import DifferentiableCurve
+from sage.manifolds.differentiable.integrated_curve import IntegratedCurve
+from sage.manifolds.differentiable.integrated_curve import IntegratedAutoparallelCurve
+from sage.manifolds.differentiable.integrated_curve import IntegratedGeodesic
+from sage.misc.cachefunc import cached_method
 
 class DifferentiableManifoldHomset(TopologicalManifoldHomset):
     r"""
@@ -500,3 +517,1350 @@ def _an_element_(self):
         coord_expression = {chart2: target_coord}
         return self.element_class(self, coord_expression)
 
+#******************************************************************************
+
+class IntegratedCurveSet(DifferentiableCurveSet):
+    r"""
+    Set of integrated curves in a differentiable manifold.
+
+    INPUT:
+
+    - ``domain`` --
+      :class:`~sage.manifolds.differentiable.real_line.OpenInterval`
+      open interval `I \subset \RR` with finite boundaries (domain of
+      the morphisms)
+    - ``codomain`` --
+      :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`;
+      differentiable manifold `M` (codomain of the morphisms)
+    - ``name`` -- (default: ``None``) string; name given to the set of
+      integrated curves; if ``None``, ``Hom_integrated(I, M)`` will be
+      used
+    - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to
+      denote the set of integrated curves; if ``None``,
+      `\mathrm{Hom_{integrated}}(I,M)` will be used
+
+    EXAMPLES:
+
+    This parent class needs to be imported::
+
+        sage: from sage.manifolds.differentiable.manifold_homset import IntegratedCurveSet
+
+    Integrated curves are only allowed to be defined on an interval with
+    finite bounds.
+    This forbids to define an instance of this parent class whose domain
+    has infinite bounds::
+
+        sage: M = Manifold(2, 'M')
+        sage: X.<x,y> = M.chart()
+        sage: R.<t> = RealLine()
+        sage: H = IntegratedCurveSet(R, M)
+        Traceback (most recent call last):
+        ...
+        ValueError: both boundaries of the interval defining the domain
+         of a Homset of integrated curves need to be finite
+
+    An instance whose domain is an interval with finite bounds allows to
+    build an integrated curve defined on the interval::
+
+        sage: I = R.open_interval(-1, 2)
+        sage: H = IntegratedCurveSet(I, M) ; H
+        Set of Morphisms from Real interval (-1, 2) to 2-dimensional
+         differentiable manifold M in Category of homsets of subobjects
+         of sets and topological spaces which actually are integrated
+         curves
+        sage: eqns_rhs = [1,1]
+        sage: vels = X.symbolic_velocities()
+        sage: t = var('t')
+        sage: p = M.point((3,4))
+        sage: Tp = M.tangent_space(p)
+        sage: v = Tp((1,2))
+        sage: c = H(eqns_rhs, vels, t, v, name='c') ; c
+        Integrated curve c in the 2-dimensional differentiable
+         manifold M
+
+    A "typical" element of ``H`` is a curve in ``M``::
+
+        sage: d = H.an_element(); d
+        Integrated curve in the 2-dimensional differentiable manifold M
+        sage: sys = d.system(verbose=True)
+        Curve in the 2-dimensional differentiable manifold M integrated
+         over the Real interval (-1, 2) as a solution to the following
+         system, written with respect to Chart (M, (x, y)):
+        <BLANKLINE>
+        Initial point: Point on the 2-dimensional differentiable
+         manifold M with coordinates [0, 0] with respect to Chart (M, (x, y))
+        Initial tangent vector: Tangent vector at Point on the
+         2-dimensional differentiable manifold M with components
+         [1/4, 0] with respect to Chart (M, (x, y))
+        <BLANKLINE>
+        d(x)/dt = Dx
+        d(y)/dt = Dy
+        d(Dx)/dt = -1/4*sin(t + 1)
+        d(Dy)/dt = 0
+        <BLANKLINE>
+
+    The test suite is passed::
+
+        sage: TestSuite(H).run()
+
+    More generally, an instance of this class may be defined with
+    abstract bounds `(a,b)`::
+
+        sage: [a,b] = var('a b')
+        sage: J = R.open_interval(a, b)
+        sage: H = IntegratedCurveSet(J, M) ; H
+        Set of Morphisms from Real interval (a, b) to 2-dimensional
+         differentiable manifold M in Category of homsets of subobjects
+         of sets and topological spaces which actually are integrated
+         curves
+
+    A "typical" element of ``H`` is a curve in ``M``::
+
+        sage: f = H.an_element(); f
+        Integrated curve in the 2-dimensional differentiable manifold M
+        sage: sys = f.system(verbose=True)
+        Curve in the 2-dimensional differentiable manifold M integrated
+         over the Real interval (a, b) as a solution to the following
+         system, written with respect to Chart (M, (x, y)):
+        <BLANKLINE>
+        Initial point: Point on the 2-dimensional differentiable
+         manifold M with coordinates [0, 0] with respect to Chart (M, (x, y))
+        Initial tangent vector: Tangent vector at Point on the
+         2-dimensional differentiable manifold M with components
+         [1/4, 0] with respect to Chart (M, (x, y))
+        <BLANKLINE>
+        d(x)/dt = Dx
+        d(y)/dt = Dy
+        d(Dx)/dt = -1/4*sin(-a + t)
+        d(Dy)/dt = 0
+        <BLANKLINE>
+
+    Yet, even in the case of abstract bounds, considering any of them to
+    be infinite is still prohibited since no numerical integration could
+    be performed::
+
+        sage: f.solve(parameters_values={a:-1, b:+oo})
+        Traceback (most recent call last):
+        ...
+        ValueError: both boundaries of the interval need to be finite
+
+    The set of integrated curves `J \longrightarrow J` is a set of
+    numerical (manifold) endomorphisms::
+
+        sage: H = IntegratedCurveSet(J, J); H
+        Set of Morphisms from Real interval (a, b) to Real interval
+         (a, b) in Category of endsets of subobjects of sets and
+         topological spaces which actually are integrated curves
+        sage: H.category()
+        Category of endsets of subobjects of sets and topological spaces
+
+    It is a monoid for the law of morphism composition::
+
+        sage: H in Monoids()
+        True
+
+    Although it is a monoid, no identity map is implemented via the
+    ``one`` method of this class or any of its subclasses.
+    This is justified by the lack of relevance of the identity map
+    within the framework of this parent class and its subclasses, whose
+    purpose is mainly devoted to numerical issues (therefore, the user
+    is left free to set a numerical version of the identity if needed)::
+
+        sage: H.one()
+        Traceback (most recent call last):
+        ...
+        ValueError: the identity is not implemented for integrated
+         curves and associated subclasses
+
+    A "typical" element of the monoid::
+
+        sage: g = H.an_element() ; g
+        Integrated curve in the Real interval (a, b)
+        sage: sys = g.system(verbose=True)
+        Curve in the Real interval (a, b) integrated over the Real
+         interval (a, b) as a solution to the following system, written
+         with respect to Chart ((a, b), (t,)):
+        <BLANKLINE>
+        Initial point: Point on the Real number line R with coordinates
+         [0] with respect to Chart ((a, b), (t,))
+        Initial tangent vector: Tangent vector at Point on the Real
+         number line R with components [1/4] with respect to
+         Chart ((a, b), (t,))
+        <BLANKLINE>
+        d(t)/ds = Dt
+        d(Dt)/ds = -1/4*sin(-a + s)
+        <BLANKLINE>
+
+    The test suite is passed, tests ``_test_one`` and ``_test_prod`` being
+    skipped for reasons mentioned above::
+
+        sage: TestSuite(H).run(skip=["_test_one", "_test_prod"])
+
+    """
+
+    Element = IntegratedCurve
+
+    def __init__(self, domain, codomain, name=None, latex_name=None):
+        r"""
+        Initialize ``self``.
+
+        TESTS::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedCurveSet
+            sage: M = Manifold(3, 'M')
+            sage: X.<x,y,z> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: H = IntegratedCurveSet(R, M)
+            Traceback (most recent call last):
+            ...
+            ValueError: both boundaries of the interval defining the
+             domain of a Homset of integrated curves need to be finite
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedCurveSet(I, M) ; H
+            Set of Morphisms from Real interval (-1, 2) to 3-dimensional
+             differentiable manifold M in Category of homsets of
+             subobjects of sets and topological spaces which actually
+             are integrated curves
+            sage: TestSuite(H).run()
+            sage: H = IntegratedCurveSet(I, I); H
+            Set of Morphisms from Real interval (-1, 2) to Real interval
+             (-1, 2) in Category of endsets of subobjects of sets and
+             topological spaces which actually are integrated curves
+            sage: TestSuite(H).run(skip=["_test_one", "_test_prod"])
+
+        """
+
+        from sage.rings.infinity import Infinity
+
+        DifferentiableCurveSet.__init__(self, domain, codomain,
+                                       name=name, latex_name=latex_name)
+
+        # checking argument 'domain': 't_min' and 't_max' are only
+        # allowed to be either expressions of finite real values
+        t_min = domain.lower_bound()
+        t_max = domain.upper_bound()
+        if t_min == -Infinity or t_max == +Infinity:
+            raise ValueError("both boundaries of the interval " +
+                             "defining the domain of a Homset of " +
+                             "integrated curves need to be finite")
+
+        if name is None:
+            self._name = "Hom_integrated({},{})".format(domain._name,
+                                                         codomain._name)
+        else:
+            self._name = name
+        if latex_name is None:
+            self._latex_name = r"\mathrm{{Hom}_{integrated}}"
+            self._latex_name += r"\left({},{}\right)".format(
+                               domain._latex_name, codomain._latex_name)
+        else:
+            self._latex_name = latex_name
+
+    #### Parent methods ####
+
+    def _repr_(self):
+        """
+        TESTS::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedCurveSet
+            sage: M = Manifold(3, 'M')
+            sage: X.<x,y,z> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedCurveSet(I, M) ; H
+            Set of Morphisms from Real interval (-1, 2) to 3-dimensional
+             differentiable manifold M in Category of homsets of
+             subobjects of sets and topological spaces which actually
+             are integrated curves
+
+        """
+        description = "Set of Morphisms "
+        description += "from {} to {} in {} ".format(self._domain,
+                                        self._codomain, self.category())
+        description += "which actually are integrated curves"
+        return description
+
+
+    def _element_constructor_(self, equations_rhs, velocities,
+                 curve_parameter, initial_tangent_vector, chart=None,
+                 name=None, latex_name=None, verbose=False):
+        r"""
+        Construct an element of ``self``, i.e. an integrated curve
+        `I \to M`, where `I` is a real interval and `M` some
+        differentiable manifold.
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedCurve`
+
+        EXAMPLES::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedCurveSet
+            sage: M = Manifold(2, 'M')
+            sage: X.<x,y> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedCurveSet(I, M)
+            sage: eqns_rhs = [1,1]
+            sage: vels = X.symbolic_velocities()
+            sage: t = var('t')
+            sage: p = M.point((3,4))
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,2))
+            sage: c = H(eqns_rhs, vels, t, v, name='c') ; c
+            Integrated curve c in the 2-dimensional differentiable
+             manifold M
+
+        """
+        # Standard construction
+        return self.element_class(self, equations_rhs, velocities,
+                curve_parameter, initial_tangent_vector, chart=chart,
+                name=name, latex_name=latex_name, verbose=verbose)
+
+    def _an_element_(self):
+        r"""
+        Construct some element of ``self``.
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedCurve`
+
+        EXAMPLES::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedCurveSet
+            sage: M = Manifold(2, 'M')
+            sage: X.<x,y> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedCurveSet(I, M)
+            sage: c = H._an_element_() ; c
+            Integrated curve in the 2-dimensional differentiable
+             manifold M
+            sage: sys = c.system(verbose=True)
+            Curve in the 2-dimensional differentiable manifold M
+             integrated over the Real interval (-1, 2) as a solution to
+             the following system, written with respect to Chart (M, (x, y)):
+            <BLANKLINE>
+            Initial point: Point on the 2-dimensional differentiable
+             manifold M with coordinates [0, 0] with respect to
+             Chart (M, (x, y))
+            Initial tangent vector: Tangent vector at Point on the
+             2-dimensional differentiable manifold M with components
+             [1/4, 0] with respect to Chart (M, (x, y))
+            <BLANKLINE>
+            d(x)/dt = Dx
+            d(y)/dt = Dy
+            d(Dx)/dt = -1/4*sin(t + 1)
+            d(Dy)/dt = 0
+            <BLANKLINE>
+            sage: sol = c.solve()
+            sage: interp = c.interpolate()
+            sage: p = c(1) ; p
+            Point on the 2-dimensional differentiable manifold M
+            sage: p.coordinates()     # abs tol 1e-12
+            (0.22732435599328793, 0.0)
+            sage: H = IntegratedCurveSet(I, I)
+            sage: c = H._an_element_() ; c
+            Integrated curve in the Real interval (-1, 2)
+            sage: sys = c.system(verbose=True)
+            Curve in the Real interval (-1, 2) integrated over the Real
+             interval (-1, 2) as a solution to the following system,
+             written with respect to Chart ((-1, 2), (t,)):
+            <BLANKLINE>
+            Initial point: Point on the Real number line R with
+             coordinates [1/2] with respect to Chart ((-1, 2), (t,))
+            Initial tangent vector: Tangent vector at Point on the Real
+             number line R with components [3/8] with respect to
+             Chart ((-1, 2), (t,))
+            <BLANKLINE>
+            d(t)/ds = Dt
+            d(Dt)/ds = -3/8*sin(s + 1)
+            sage: sol = c.solve()
+            sage: interp = c.interpolate()
+            sage: p = c(1) ; p
+            Point on the Real number line R
+            sage: p.coordinates()     # abs tol 1e-12
+            (0.840986533989932,)
+
+        """
+
+        from sage.categories.homset import Hom
+        from sage.functions.trig import sin
+        from sage.symbolic.ring import var
+
+        dom = self.domain()
+        t = dom.canonical_coordinate()
+        t_min = dom.lower_bound() # this is either an expression or a
+        # finite value thanks to tests in '__init__'
+        t_max = dom.upper_bound() # this is either an expression or a
+        # finite value thanks to tests in '__init__'
+
+        codom = self.codomain()
+        dim = codom.dim()
+        chart2 = codom.default_chart()
+        # In case the codomain coincides with the domain,
+        # it is important to distinguish between the canonical
+        # coordinate, and the curve parameter since, in such a
+        # situation, the coordinate should not be used to denote the
+        # curve parameter, since it actually becomes a function of the
+        # curve parameter, and such a function is an unknown of the
+        # system defining the curve.
+        # In other cases, it might still happen for a coordinate of the
+        # codomain to be denoted the same as the canonical coordinate of
+        # the domain (for instance, the codomain could be another
+        # real interval, different from the domain, and yet with same
+        # letter denoting its canonical coordinate).
+        # In such case, an error is raised from method 'init'
+        # of class IntegratedCurve; to solve it, the user is
+        # free to change the name of the codomain coordinate in the
+        # chart used on the codomain.
+        if dom == codom:
+            param = var('s')
+            if t == param: # the canonical coordinate of the domain
+            # might be the expression 's' even though it was affected
+            # above to the variable 't'
+                param = var('u')
+        else:
+            param = t
+
+        # An analytical curve is used to find a region of the codomain
+        # where a certain integrated curve may be defined:
+        H = Hom(dom, codom)
+        c = H.an_element()
+        x0_A = c.expr()[0].substitute({t:1})
+        x0_B = c.expr()[0].substitute({t:0}) # necessarily, x0_A < x0_B
+        p_coords = [x0_A] + list(c.expr()[1:dim])
+        p = codom.point(p_coords)
+
+        # The initial tangent vector:
+        v_comps = [(x0_B-x0_A)/2] + [0 for i in range(dim-1)]
+        v = codom.tangent_space(p)(v_comps)
+
+        # The equations defining the curve:
+        eqns_rhs=[-(x0_B-x0_A)/2*sin(param-t_min)]+[0 for i in range(dim-1)]
+        # combined with the initial components above, all velocities
+        # vanish, except the first one, which is a cosine function.
+        # This differential system results in a curve constant in all
+        # its coordinates, except the first one, which oscillates around
+        # the value 'x0_A' with an amplitude '(x0_B-x0_A)/2'
+
+        # The symbolic expressions for the velocities:
+        vels = chart2.symbolic_velocities()
+
+        return self.element_class(self,eqns_rhs,vels,param,v)
+
+    def one(self):
+        r"""
+        Raise an error refusing to provide the identity element.
+        This overrides the ``one`` method of class
+        :class:`~sage.manifolds.manifold_homset.TopologicalManifoldHomset`,
+        which would actually raise an error as well, due to lack of option
+        ``is_identity`` in ``element_constructor`` method of ``self``.
+
+        TESTS::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedCurveSet
+            sage: M = Manifold(3, 'M')
+            sage: X.<x,y,z> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedCurveSet(I, M)
+            sage: H.one()
+            Traceback (most recent call last):
+            ...
+            TypeError: Set of Morphisms from Real interval (-1, 2) to
+             3-dimensional differentiable manifold M in Category of
+             homsets of subobjects of sets and topological spaces which
+             actually are integrated curves is not a monoid
+            sage: H = IntegratedCurveSet(I, I)
+            sage: H.one()
+            Traceback (most recent call last):
+            ...
+            ValueError: the identity is not implemented for integrated
+             curves and associated subclasses
+
+        """
+
+        if self.codomain() != self.domain():
+            raise TypeError("{} is not a monoid".format(self))
+        else:
+            raise ValueError("the identity is not implemented for " +
+                            "integrated curves and associated " +
+                            "subclasses")
+
+#******************************************************************************
+
+class IntegratedAutoparallelCurveSet(IntegratedCurveSet):
+    r"""
+    Set of integrated autoparallel curves in a differentiable manifold.
+
+    INPUT:
+
+    - ``domain`` --
+      :class:`~sage.manifolds.differentiable.real_line.OpenInterval`
+      open interval `I \subset \RR` with finite boundaries (domain of
+      the morphisms)
+    - ``codomain`` --
+      :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`;
+      differentiable manifold `M` (codomain of the morphisms)
+    - ``name`` -- (default: ``None``) string; name given to the set of
+      integrated autoparallel curves; if ``None``,
+      ``Hom_autoparallel(I, M)`` will be used
+    - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote
+      the set of integrated autoparallel curves; if ``None``,
+      `\mathrm{Hom_{autoparallel}}(I,M)` will be used
+
+    EXAMPLES:
+
+    This parent class needs to be imported::
+
+        sage: from sage.manifolds.differentiable.manifold_homset import IntegratedAutoparallelCurveSet
+
+    Integrated autoparallel curves are only allowed to be defined on an
+    interval with finite bounds.
+    This forbids to define an instance of this parent class whose domain
+    has infinite bounds::
+
+        sage: M = Manifold(2, 'M')
+        sage: X.<x,y> = M.chart()
+        sage: R.<t> = RealLine()
+        sage: H = IntegratedAutoparallelCurveSet(R, M)
+        Traceback (most recent call last):
+        ...
+        ValueError: both boundaries of the interval defining the domain
+         of a Homset of integrated autoparallel curves need to be finite
+
+    An instance whose domain is an interval with finite bounds allows to
+    build a curve that is autoparallel with respect to a connection
+    defined on the codomain::
+
+        sage: I = R.open_interval(-1, 2)
+        sage: H = IntegratedAutoparallelCurveSet(I, M) ; H
+        Set of Morphisms from Real interval (-1, 2) to 2-dimensional
+         differentiable manifold M in Category of homsets of subobjects
+         of sets and topological spaces which actually are integrated
+         autoparallel curves with respect to a certain affine connection
+        sage: nab = M.affine_connection('nabla')
+        sage: nab[0,1,0], nab[0,0,1] = 1,2
+        sage: nab.torsion()[:]
+        [[[0, -1], [1, 0]], [[0, 0], [0, 0]]]
+        sage: t = var('t')
+        sage: p = M.point((3,4))
+        sage: Tp = M.tangent_space(p)
+        sage: v = Tp((1,2))
+        sage: c = H(nab, t, v, name='c') ; c
+        Integrated autoparallel curve c in the 2-dimensional
+         differentiable manifold M
+
+    A "typical" element of ``H`` is an autoparallel curve in ``M``::
+
+        sage: d = H.an_element(); d
+        Integrated autoparallel curve in the 2-dimensional
+         differentiable manifold M
+        sage: sys = d.system(verbose=True)
+        Autoparallel curve in the 2-dimensional differentiable manifold
+         M equipped with Affine connection nab on the 2-dimensional
+         differentiable manifold M, and integrated over the Real
+         interval (-1, 2) as a solution to the following equations,
+         written with respect to Chart (M, (x, y)):
+        <BLANKLINE>
+        Initial point: Point on the 2-dimensional differentiable
+         manifold M with coordinates [0, -1/2] with respect to
+         Chart (M, (x, y))
+        Initial tangent vector: Tangent vector at Point on the
+         2-dimensional differentiable manifold M with components
+         [-1/6/(e^(-1) - 1), 1/3] with respect to Chart (M, (x, y))
+        <BLANKLINE>
+        d(x)/dt = Dx
+        d(y)/dt = Dy
+        d(Dx)/dt = -Dx*Dy
+        d(Dy)/dt = 0
+        <BLANKLINE>
+
+    The test suite is passed::
+
+        sage: TestSuite(H).run()
+
+    For any open interval `J` with finite bounds `(a,b)`, all curves are
+    autoparallel with respect to any connection.
+    Therefore, the set of autoparallel curves `J \longrightarrow J` is a
+    set of numerical (manifold) endomorphisms that is a monoid for the
+    law of morphism composition::
+
+        sage: [a,b] = var('a b')
+        sage: J = R.open_interval(a, b)
+        sage: H = IntegratedAutoparallelCurveSet(J, J); H
+        Set of Morphisms from Real interval (a, b) to Real interval
+         (a, b) in Category of endsets of subobjects of sets and
+         topological spaces which actually are integrated autoparallel
+         curves with respect to a certain affine connection
+        sage: H.category()
+        Category of endsets of subobjects of sets and topological spaces
+        sage: H in Monoids()
+        True
+
+    Although it is a monoid, no identity map is implemented via the
+    ``one`` method of this class or its subclass devoted to geodesics.
+    This is justified by the lack of relevance of the identity map
+    within the framework of this parent class and its subclass, whose
+    purpose is mainly devoted to numerical issues (therefore, the user
+    is left free to set a numerical version of the identity if needed)::
+
+        sage: H.one()
+        Traceback (most recent call last):
+        ...
+        ValueError: the identity is not implemented for integrated
+         curves and associated subclasses
+
+    A "typical" element of the monoid::
+
+        sage: g = H.an_element() ; g
+        Integrated autoparallel curve in the Real interval (a, b)
+        sage: sys = g.system(verbose=True)
+        Autoparallel curve in the Real interval (a, b) equipped with
+         Affine connection nab on the Real interval (a, b), and
+         integrated over the Real interval (a, b) as a solution to the
+         following equations, written with respect to Chart ((a, b), (t,)):
+        <BLANKLINE>
+        Initial point: Point on the Real number line R with coordinates
+         [0] with respect to Chart ((a, b), (t,))
+        Initial tangent vector: Tangent vector at Point on the Real
+         number line R with components
+         [-(e^(1/2) - 1)/(a - b)] with respect to
+         Chart ((a, b), (t,))
+        <BLANKLINE>
+        d(t)/ds = Dt
+        d(Dt)/ds = -Dt^2
+        <BLANKLINE>
+
+    The test suite is passed, tests ``_test_one`` and ``_test_prod`` being
+    skipped for reasons mentioned above::
+
+        sage: TestSuite(H).run(skip=["_test_one", "_test_prod"])
+
+    """
+
+    Element = IntegratedAutoparallelCurve
+
+    def __init__(self, domain, codomain, name=None, latex_name=None):
+        r"""
+        Initialize ``self``.
+
+        TESTS::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedAutoparallelCurveSet
+            sage: M = Manifold(3, 'M')
+            sage: X.<x,y,z> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: H = IntegratedAutoparallelCurveSet(R, M)
+            Traceback (most recent call last):
+            ...
+            ValueError: both boundaries of the interval defining the
+             domain of a Homset of integrated autoparallel curves need
+             to be finite
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedAutoparallelCurveSet(I, M) ; H
+            Set of Morphisms from Real interval (-1, 2) to 3-dimensional
+             differentiable manifold M in Category of homsets of
+             subobjects of sets and topological spaces which actually are
+             integrated autoparallel curves with respect to a certain
+             affine connection
+            sage: TestSuite(H).run()
+            sage: H = IntegratedAutoparallelCurveSet(I, I); H
+            Set of Morphisms from Real interval (-1, 2) to Real interval
+             (-1, 2) in Category of endsets of subobjects of sets and
+             topological spaces which actually are integrated
+             autoparallel curves with respect to a certain affine connection
+            sage: TestSuite(H).run(skip=["_test_one", "_test_prod"])
+
+        """
+
+        from sage.rings.infinity import Infinity
+
+        DifferentiableCurveSet.__init__(self, domain, codomain,
+                                       name=name, latex_name=latex_name)
+
+        # checking argument 'domain'
+        t_min = domain.lower_bound()
+        t_max = domain.upper_bound()
+        if t_min == -Infinity or t_max == +Infinity:
+            raise ValueError("both boundaries of the interval " +
+                             "defining the domain of a Homset of " +
+                             "integrated autoparallel curves need to " +
+                             "be finite")
+
+        if name is None:
+            self._name = "Hom_autoparallel"
+            self._name += "({},{})".format(domain._name, codomain._name)
+        else:
+            self._name = name
+        if latex_name is None:
+            self._latex_name=r"\mathrm{{Hom}_{autoparallel}}"
+            self._latex_name+= r"\left({},{}\right)".format(
+                               domain._latex_name, codomain._latex_name)
+        else:
+            self._latex_name = latex_name
+
+    #### Parent methods ####
+
+    def _repr_(self):
+        """
+        TESTS::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedAutoparallelCurveSet
+            sage: M = Manifold(3, 'M')
+            sage: X.<x,y,z> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedAutoparallelCurveSet(I, M) ; H
+            Set of Morphisms from Real interval (-1, 2) to 3-dimensional
+             differentiable manifold M in Category of homsets of
+             subobjects of sets and topological spaces which actually
+             are integrated autoparallel curves with respect to a certain
+             affine connection
+
+        """
+
+        description = "Set of Morphisms "
+        description += "from {} to {} in {} ".format(self._domain,
+                                        self._codomain, self.category())
+        description += "which actually are integrated autoparallel "
+        description += "curves with respect to a certain affine connection"
+        return description
+
+
+    def _element_constructor_(self, affine_connection, curve_parameter,
+                    initial_tangent_vector, chart=None, name=None,
+                    latex_name=None, verbose=False):
+        r"""
+        Construct an element of ``self``, i.e. an integrated
+        autoparallel curve `I \to M`, where `I` is a real interval and
+        `M` some differentiable manifold.
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedAutoparallelCurve`
+
+        EXAMPLES::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedAutoparallelCurveSet
+            sage: M = Manifold(2, 'M')
+            sage: X.<x,y> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedAutoparallelCurveSet(I, M)
+            sage: nab = M.affine_connection('nabla')
+            sage: nab[0,1,0], nab[0,0,1] = 1,2
+            sage: nab.torsion()[:]
+            [[[0, -1], [1, 0]], [[0, 0], [0, 0]]]
+            sage: t = var('t')
+            sage: p = M.point((3,4))
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,2))
+            sage: c = H(nab, t, v, name='c') ; c
+            Integrated autoparallel curve c in the 2-dimensional
+             differentiable manifold M
+
+        """
+        # Standard construction
+        return self.element_class(self, affine_connection,
+                 curve_parameter, initial_tangent_vector, chart=chart,
+                 name=name,latex_name=latex_name, verbose=verbose)
+
+    def _an_element_(self):
+        r"""
+        Construct some element of ``self``.
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedAutoparallelCurve`
+
+        EXAMPLES::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedAutoparallelCurveSet
+            sage: M = Manifold(2, 'M')
+            sage: X.<x,y> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: [a,b] = var('a b')
+            sage: J = R.open_interval(a, b)
+            sage: H = IntegratedAutoparallelCurveSet(J, M)
+            sage: c = H._an_element_() ; c
+            Integrated autoparallel curve in the 2-dimensional
+             differentiable manifold M
+            sage: sys = c.system(verbose=True)
+            Autoparallel curve in the 2-dimensional differentiable
+             manifold M equipped with Affine connection nab on the
+             2-dimensional differentiable manifold M, and integrated
+             over the Real interval (a, b) as a solution to the
+             following equations, written with respect to Chart (M, (x, y)):
+            <BLANKLINE>
+            Initial point: Point on the 2-dimensional differentiable
+             manifold M with coordinates [0, -1/2]
+             with respect to Chart (M, (x, y))
+            Initial tangent vector: Tangent vector at Point on the
+             2-dimensional differentiable manifold M with components
+             [1/2/((a - b)*(e^(-1) - 1)), -1/(a - b)] with respect to
+             Chart (M, (x, y))
+            <BLANKLINE>
+            d(x)/dt = Dx
+            d(y)/dt = Dy
+            d(Dx)/dt = -Dx*Dy
+            d(Dy)/dt = 0
+            <BLANKLINE>
+            sage: sol = c.solve(parameters_values={a:0,b:4})
+            sage: interp = c.interpolate()
+            sage: p = c(1) ; p
+            Point on the 2-dimensional differentiable manifold M
+            sage: p.coordinates()     # abs tol 1e-12
+            (0.1749660043664451, -0.2499999999999998)
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedAutoparallelCurveSet(I, I)
+            sage: c = H._an_element_() ; c
+            Integrated autoparallel curve in the Real interval (-1, 2)
+            sage: sys = c.system(verbose=True)
+            Autoparallel curve in the Real interval (-1, 2) equipped
+             with Affine connection nab on the Real interval (-1, 2),
+             and integrated over the Real interval (-1, 2) as a solution
+             to the following equations, written with respect to
+             Chart ((-1, 2), (t,)):
+            <BLANKLINE>
+            Initial point: Point on the Real number line R with
+             coordinates [1/2] with respect to Chart ((-1, 2), (t,))
+            Initial tangent vector: Tangent vector at Point on the Real
+             number line R with components [1/3*e^(3/4) - 1/3]
+             with respect to Chart ((-1, 2), (t,))
+            <BLANKLINE>
+            d(t)/ds = Dt
+            d(Dt)/ds = -Dt^2
+            <BLANKLINE>
+            sage: sol = c.solve()
+            sage: interp = c.interpolate()
+            sage: p = c(1) ; p
+            Point on the Real number line R
+            sage: p.coordinates()     # abs tol 1e-12
+            (1.0565635215890166,)
+
+        """
+
+        from sage.rings.infinity import Infinity
+        from sage.rings.rational_field import QQ
+        from sage.categories.homset import Hom
+        from sage.symbolic.ring import var
+        from sage.functions.log import exp
+
+        dom = self.domain()
+        t = dom.canonical_coordinate()
+        t_min = dom.lower_bound() # this is either an expression or a
+        # finite value thanks to tests in '__init__'
+        t_max = dom.upper_bound() # idem
+
+        codom = self.codomain()
+        dim = codom.dim()
+        i0 = codom.start_index()
+        chart2 = codom.default_chart()
+        x = chart2[:][0]
+        # In case the codomain coincides with the domain,
+        # it is important to distinguish between the canonical
+        # coordinate, and the curve parameter since, in such a
+        # situation, the coordinate should not be used to denote the
+        # curve parameter, since it actually becomes a function of the
+        # curve parameter, and such a function is an unknown of the
+        # system defining the curve.
+        # In other cases, it might still happen for a coordinate of the
+        # codomain to be denoted the same as the canonical coordinate of
+        # the domain (for instance, the codomain could be another
+        # real interval, different from the domain, and yet with same
+        # letter denoting its canonical coordinate).
+        # In such case, an error is raised from method 'init'
+        # of class IntegratedCurve; to solve it, the user is
+        # free to change the name of the codomain coordinate in the
+        # chart used on the codomain.
+        if dom == codom:
+            param = var('s')
+        else:
+            param = t
+
+        # An analytical curve is used to find a region of the codomain
+        # where a certain integrated autoparallel curve may be defined:
+        H = Hom(dom, codom)
+        c = H.an_element()
+        x_A = c.expr()[0].substitute({t:1})
+        x_B = c.expr()[0].substitute({t:0}) # necessarily, x_A < x_B
+
+        if dim == 1:
+            nab = codom.affine_connection('nab')
+            nab.set_coef()[i0,i0,i0] = 1
+
+            # The initial point:
+            p = codom.point([x_A])
+
+            # The initial tangent vector:
+            x_dot_A = (exp(x_B - x_A) - 1)/(t_max - t_min)
+            v = codom.tangent_space(p)([x_dot_A])
+
+            return self.element_class(self, nab, param, v)
+            # the autoparallel curve returned will correspond to the
+            # following analytical solution:
+            # x(t) = ln( x_dot_A*(t-t_min) + 1 ) + x_A, which is such
+            # that x(t_min) = x_A and x(t_max) = x_B due to x_dot_A
+            # set to the value above
+
+        # else: (i.e. dim >= 2)
+
+        nab = codom.affine_connection('nab')
+        nab.set_coef()[i0,i0,i0+1] = 1
+
+        y_bounds = chart2._bounds[1]  # bounds of second coordinate
+        # Determination of an interval (y_A, y_B) arround target_point:
+        y_min = y_bounds[0][0]
+        y_max = y_bounds[1][0]
+        one_half = QQ(1) / QQ(2)
+        if y_min == -Infinity:
+            if y_max == Infinity:
+                y_A = - one_half
+                y_B = one_half
+            else:
+                y_A = y_max - 3*one_half
+                y_B = y_max - one_half
+        else:
+            if y_max == Infinity:
+                y_A = y_min + one_half
+                y_B = y_min + 3*one_half
+            else:
+                dy = (y_max - y_min) / 4
+                y_A = y_min + dy
+                y_B = y_max - dy
+
+        # The initial point:
+        p_coords = [x_A] + [y_A] + list(c.expr()[2:dim])
+        p = codom.point(p_coords)
+
+        # The initial tangent vector:
+        y_dot_A = (y_B - y_A) / (t_max - t_min)
+        x_dot_A = y_dot_A*(x_B - x_A) / (1-exp(-y_dot_A*(t_max-t_min)))
+        v_comps = [x_dot_A] + [y_dot_A] + [0 for i in range(dim-2)]
+        v = codom.tangent_space(p)(v_comps)
+
+        return self.element_class(self, nab, param, v)
+        # the autoparallel curve returned will correspond to the
+        # following analytical solution:
+        # all coordinates other than the first two coordinates are
+        # constant, and
+        # x(t) = x_dot_A/y_dot_A*(1 - exp(-y_dot_A*(t-t_min))) + x_A
+        # y(t) = y_dot_A*(t-t_min) + y_A
+        # This solution is such that
+        # x(t_min) = x_A and x(t_max) = x_B due to x_dot_A set to the
+        # value above, and
+        # y(t_min) = y_A and y(t_max) = y_B due to y_dot_A set to the
+        # value above
+
+#******************************************************************************
+
+class IntegratedGeodesicSet(IntegratedAutoparallelCurveSet):
+    r"""
+    Set of integrated geodesic in a differentiable manifold.
+
+    INPUT:
+
+    - ``domain`` --
+      :class:`~sage.manifolds.differentiable.real_line.OpenInterval`
+      open interval `I \subset \RR` with finite boundaries (domain of
+      the morphisms)
+    - ``codomain`` --
+      :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`;
+      differentiable manifold `M` (codomain of the morphisms)
+    - ``name`` -- (default: ``None``) string; name given to the set of
+      integrated geodesics; if ``None``, ``Hom_geodesic(I, M)`` will be used
+    - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote
+      the set of integrated geodesics; if ``None``,
+      `\mathrm{Hom_{geodesic}}(I,M)` will be used
+
+    EXAMPLES:
+
+    This parent class needs to be imported::
+
+        sage: from sage.manifolds.differentiable.manifold_homset import IntegratedGeodesicSet
+
+    Integrated geodesics are only allowed to be defined on an interval
+    with finite bounds.
+    This forbids to define an instance of this parent class whose domain
+    has infinite bounds::
+
+        sage: M = Manifold(2, 'M')
+        sage: X.<x,y> = M.chart()
+        sage: R.<t> = RealLine()
+        sage: H = IntegratedGeodesicSet(R, M)
+        Traceback (most recent call last):
+        ...
+        ValueError: both boundaries of the interval defining the domain
+         of a Homset of integrated geodesics need to be finite
+
+    An instance whose domain is an interval with finite bounds allows to
+    build a geodesic with respect to a metric defined on the codomain::
+
+        sage: I = R.open_interval(-1, 2)
+        sage: H = IntegratedGeodesicSet(I, M) ; H
+        Set of Morphisms from Real interval (-1, 2) to 2-dimensional
+         differentiable manifold M in Category of homsets of subobjects
+         of sets and topological spaces which actually are integrated
+         geodesics with respect to a certain metric
+        sage: g = M.metric('g')
+        sage: g[0,0], g[1,1], g[0,1] = 1, 1, 2
+        sage: t = var('t')
+        sage: p = M.point((3,4))
+        sage: Tp = M.tangent_space(p)
+        sage: v = Tp((1,2))
+        sage: c = H(g, t, v, name='c') ; c
+        Integrated geodesic c in the 2-dimensional differentiable
+         manifold M
+
+    A "typical" element of ``H`` is a geodesic in ``M``::
+
+        sage: d = H.an_element(); d
+        Integrated geodesic in the 2-dimensional differentiable
+         manifold M
+        sage: sys = d.system(verbose=True)
+        Geodesic in the 2-dimensional differentiable manifold M equipped
+         with Riemannian metric g on the 2-dimensional differentiable
+         manifold M, and integrated over the Real interval (-1, 2) as a
+         solution to the following geodesic equations, written
+         with respect to Chart (M, (x, y)):
+        <BLANKLINE>
+        Initial point: Point on the 2-dimensional differentiable
+         manifold M with coordinates [0, 0] with respect to
+         Chart (M, (x, y))
+        Initial tangent vector: Tangent vector at Point on the
+         2-dimensional differentiable manifold M with components
+         [1/3*e^(1/2) - 1/3, 0] with respect to Chart (M, (x, y))
+        <BLANKLINE>
+        d(x)/dt = Dx
+        d(y)/dt = Dy
+        d(Dx)/dt = -Dx^2
+        d(Dy)/dt = 0
+
+    The test suite is passed::
+
+        sage: TestSuite(H).run()
+
+    For any open interval `J` with finite bounds `(a,b)`, all curves are
+    geodesics with respect to any metric.
+    Therefore, the set of geodesics `J \longrightarrow J` is a set of
+    numerical (manifold) endomorphisms that is a monoid for the law of
+    morphism composition::
+
+        sage: [a,b] = var('a b')
+        sage: J = R.open_interval(a, b)
+        sage: H = IntegratedGeodesicSet(J, J); H
+        Set of Morphisms from Real interval (a, b) to Real interval
+         (a, b) in Category of endsets of subobjects of sets and
+         topological spaces which actually are integrated geodesics
+         with respect to a certain metric
+        sage: H.category()
+        Category of endsets of subobjects of sets and topological spaces
+        sage: H in Monoids()
+        True
+
+    Although it is a monoid, no identity map is implemented via the
+    ``one`` method of this class.
+    This is justified by the lack of relevance of the identity map
+    within the framework of this parent class, whose purpose is mainly
+    devoted to numerical issues (therefore, the user is left free to set
+    a numerical version of the identity if needed)::
+
+        sage: H.one()
+        Traceback (most recent call last):
+        ...
+        ValueError: the identity is not implemented for integrated
+         curves and associated subclasses
+
+    A "typical" element of the monoid::
+
+        sage: g = H.an_element() ; g
+        Integrated geodesic in the Real interval (a, b)
+        sage: sys = g.system(verbose=True)
+        Geodesic in the Real interval (a, b) equipped with Riemannian
+         metric g on the Real interval (a, b), and integrated over the
+         Real interval (a, b) as a solution to the following geodesic
+         equations, written with respect to Chart ((a, b), (t,)):
+        <BLANKLINE>
+        Initial point: Point on the Real number line R with coordinates
+         [0] with respect to Chart ((a, b), (t,))
+        Initial tangent vector: Tangent vector at Point on the Real
+         number line R with components [-(e^(1/2) - 1)/(a - b)]
+         with respect to Chart ((a, b), (t,))
+        <BLANKLINE>
+        d(t)/ds = Dt
+        d(Dt)/ds = -Dt^2
+        <BLANKLINE>
+
+    The test suite is passed, tests ``_test_one`` and ``_test_prod`` being
+    skipped for reasons mentioned above::
+
+        sage: TestSuite(H).run(skip=["_test_one", "_test_prod"])
+
+    """
+
+    Element = IntegratedGeodesic
+
+    def __init__(self, domain, codomain, name=None, latex_name=None):
+        r"""
+        Initialize ``self``.
+
+        TESTS::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedGeodesicSet
+            sage: M = Manifold(3, 'M')
+            sage: X.<x,y,z> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: H = IntegratedGeodesicSet(R, M)
+            Traceback (most recent call last):
+            ...
+            ValueError: both boundaries of the interval defining the
+             domain of a Homset of integrated geodesics need to be
+             finite
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedGeodesicSet(I, M) ; H
+            Set of Morphisms from Real interval (-1, 2) to 3-dimensional
+             differentiable manifold M in Category of homsets of
+             subobjects of sets and topological spaces which actually
+             are integrated geodesics with respect to a certain metric
+            sage: TestSuite(H).run()
+            sage: H = IntegratedGeodesicSet(I, I); H
+            Set of Morphisms from Real interval (-1, 2) to Real interval
+             (-1, 2) in Category of endsets of subobjects of sets and
+             topological spaces which actually are integrated geodesics
+             with respect to a certain metric
+            sage: TestSuite(H).run(skip=["_test_one", "_test_prod"])
+
+        """
+
+        from sage.rings.infinity import Infinity
+
+        DifferentiableCurveSet.__init__(self, domain, codomain,
+                                       name=name, latex_name=latex_name)
+
+        # checking argument 'domain'
+        t_min = domain.lower_bound()
+        t_max = domain.upper_bound()
+        if t_min == -Infinity or t_max == +Infinity:
+            raise ValueError("both boundaries of the interval " +
+                             "defining the domain of a Homset of " +
+                             "integrated geodesics need to be finite")
+
+        if name is None:
+            self._name = "Hom_geodesic"
+            self._name += "({},{})".format(domain._name, codomain._name)
+        else:
+            self._name = name
+        if latex_name is None:
+            self._latex_name = r"\mathrm{{Hom}_{geodesic}}"
+            self._latex_name+= r"\left({},{}\right)".format(
+                               domain._latex_name, codomain._latex_name)
+        else:
+            self._latex_name = latex_name
+
+    #### Parent methods ####
+
+    def _repr_(self):
+        """
+        TESTS::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedGeodesicSet
+            sage: M = Manifold(3, 'M')
+            sage: X.<x,y,z> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedGeodesicSet(I, M) ; H
+            Set of Morphisms from Real interval (-1, 2) to 3-dimensional
+             differentiable manifold M in Category of homsets of
+             subobjects of sets and topological spaces which actually
+             are integrated geodesics with respect to a certain metric
+
+        """
+        description = "Set of Morphisms "
+        description += "from {} to {} in {} ".format(self._domain,
+                                        self._codomain, self.category())
+        description += "which actually are integrated geodesics "
+        description += "with respect to a certain metric"
+        return description
+
+    def _element_constructor_(self, metric, curve_parameter,
+                    initial_tangent_vector, chart=None, name=None,
+                    latex_name=None, verbose=False):
+        r"""
+        Construct an element of ``self``, i.e. an integrated geodesic
+        `I \to M`, where `I` is a real interval and
+        `M` some differentiable manifold.
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedGeodesic`
+
+        EXAMPLES::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedGeodesicSet
+            sage: M = Manifold(2, 'M')
+            sage: X.<x,y> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedGeodesicSet(I, M)
+            sage: g = M.metric('g')
+            sage: g[0,0], g[1,1], g[0,1] = 1, 1, 2
+            sage: t = var('t')
+            sage: p = M.point((3,4))
+            sage: Tp = M.tangent_space(p)
+            sage: v = Tp((1,2))
+            sage: c = H(g, t, v, name='c') ; c
+            Integrated geodesic c in the 2-dimensional differentiable
+             manifold M
+
+        """
+        # Standard construction
+        return self.element_class(self, metric, curve_parameter,
+                 initial_tangent_vector, chart=chart, name=name,
+                 latex_name=latex_name, verbose=verbose)
+
+    def _an_element_(self):
+        r"""
+        Construct some element of ``self``.
+
+        OUTPUT:
+
+        - :class:`~sage.manifolds.differentiable.integrated_curve.IntegratedGeodesic`
+
+        EXAMPLES::
+
+            sage: from sage.manifolds.differentiable.manifold_homset import IntegratedGeodesicSet
+            sage: M = Manifold(4, 'M', start_index=1)
+            sage: X.<w,x,y,z> = M.chart()
+            sage: R.<t> = RealLine()
+            sage: [a,b] = var('a b')
+            sage: J = R.open_interval(a, b)
+            sage: H = IntegratedGeodesicSet(J, M)
+            sage: c = H._an_element_() ; c
+            Integrated geodesic in the 4-dimensional differentiable
+             manifold M
+            sage: sys = c.system(verbose=True)
+            Geodesic in the 4-dimensional differentiable manifold M
+             equipped with Riemannian metric g on the 4-dimensional
+             differentiable manifold M, and integrated over the Real
+             interval (a, b) as a solution to the following geodesic
+             equations, written with respect to Chart (M, (w, x, y, z)):
+            <BLANKLINE>
+            Initial point: Point on the 4-dimensional differentiable
+             manifold M with coordinates [0, 0, 0, 0] with respect to
+             Chart (M, (w, x, y, z))
+            Initial tangent vector: Tangent vector at Point on the
+             4-dimensional differentiable manifold M with components
+             [-(e^(1/2) - 1)/(a - b), 0, 0, 0] with respect to
+             Chart (M, (w, x, y, z))
+            <BLANKLINE>
+            d(w)/dt = Dw
+            d(x)/dt = Dx
+            d(y)/dt = Dy
+            d(z)/dt = Dz
+            d(Dw)/dt = -Dw^2
+            d(Dx)/dt = 0
+            d(Dy)/dt = 0
+            d(Dz)/dt = 0
+            <BLANKLINE>
+            sage: sol = c.solve(parameters_values={a:1,b:6})
+            sage: interp = c.interpolate()
+            sage: p = c(3) ; p
+            Point on the 4-dimensional differentiable manifold M
+            sage: p.coordinates()     # abs tol 1e-12
+            (0.2307056927167852, 0.0, 0.0, 0.0)
+            sage: I = R.open_interval(-1, 2)
+            sage: H = IntegratedGeodesicSet(I, I)
+            sage: c = H._an_element_() ; c
+            Integrated geodesic in the Real interval (-1, 2)
+            sage: sys = c.system(verbose=True)
+            Geodesic in the Real interval (-1, 2) equipped with
+             Riemannian metric g on the Real interval (-1, 2), and
+             integrated over the Real interval (-1, 2) as a solution to
+             the following geodesic equations, written with respect to
+             Chart ((-1, 2), (t,)):
+            <BLANKLINE>
+            Initial point: Point on the Real number line R with
+             coordinates [1/2] with respect to Chart ((-1, 2), (t,))
+            Initial tangent vector: Tangent vector at Point on the Real
+             number line R with components [1/3*e^(3/4) - 1/3]
+             with respect to Chart ((-1, 2), (t,))
+            <BLANKLINE>
+            d(t)/ds = Dt
+            d(Dt)/ds = -Dt^2
+            <BLANKLINE>
+            sage: sol = c.solve()
+            sage: interp = c.interpolate()
+            sage: p = c(1) ; p
+            Point on the Real number line R
+            sage: p.coordinates()     # abs tol 1e-12
+            (1.0565635215890166,)
+
+        """
+
+        from sage.categories.homset import Hom
+        from sage.symbolic.ring import var
+        from sage.functions.log import exp
+
+        dom = self.domain()
+        t = dom.canonical_coordinate()
+        t_min = dom.lower_bound() # this is either an expression or a
+        # finite value thanks to tests in '__init__'
+        t_max = dom.upper_bound() # idem
+
+        codom = self.codomain()
+        dim = codom.dim()
+        i0 = codom.start_index()
+        chart2 = codom.default_chart()
+        x = chart2[:][0]
+        # In case the codomain coincides with the domain,
+        # it is important to distinguish between the canonical
+        # coordinate, and the curve parameter since, in such a
+        # situation, the coordinate should not be used to denote the
+        # curve parameter, since it actually becomes a function of the
+        # curve parameter, and such a function is an unknown of the
+        # system defining the curve.
+        # In other cases, it might still happen for a coordinate of the
+        # codomain to be denoted the same as the canonical coordinate of
+        # the domain (for instance, the codomain could be another
+        # real interval, different from the domain, and yet with same
+        # letter denoting its canonical coordinate).
+        # In such case, an error is raised from method 'init'
+        # of class IntegratedCurve; to solve it, the user is
+        # free to change the name of the codomain coordinate in the
+        # chart used on the codomain.
+        if dom == codom:
+            param = var('s')
+        else:
+            param = t
+
+        # An analytical curve is used to find a region of the codomain
+        # where a certain integrated autoparallel curve may be defined:
+        H = Hom(dom, codom)
+        c = H.an_element()
+        x_A = c.expr()[0].substitute({t:1})
+        x_B = c.expr()[0].substitute({t:0}) # necessarily, x_A < x_B
+
+        g = codom.metric('g')
+        g[i0,i0] = exp(2*x)
+        if dim > 1:
+            for i in range(1,dim):
+                g[i0+i,i0+i] = 1
+
+        # The initial point:
+        p_coords = [x_A] + list(c.expr()[1:dim])
+        p = codom.point(p_coords)
+
+        # The initial tangent vector:
+        x_dot_A = (exp(x_B - x_A) - 1)/(t_max - t_min)
+        v_comps = [x_dot_A] + [0 for i in range(dim-1)]
+        v = codom.tangent_space(p)(v_comps)
+
+        return self.element_class(self, g, param, v)
+        # the geodesic returned will correspond to the following
+        # analytical solution:
+        # all coordinates other than the first one are constant, and
+        # x(t) = ln( x_dot_A*(t-t_min) + 1 ) + x_A, which is such that
+        # x(t_min) = x_A and x(t_max) = x_B due to x_dot_A set to the
+        # value above
diff --git a/src/sage/manifolds/differentiable/tensorfield.py b/src/sage/manifolds/differentiable/tensorfield.py
index e2ba35c4af0..c0343953fb6 100644
--- a/src/sage/manifolds/differentiable/tensorfield.py
+++ b/src/sage/manifolds/differentiable/tensorfield.py
@@ -936,7 +936,7 @@ def add_comp(self, basis=None):
         for assignment.
 
         The components with respect to other frames having the same domain
-        as the provided vector frame are kept. To delete them them, use the
+        as the provided vector frame are kept. To delete them, use the
         method :meth:`set_comp` instead.
 
         INPUT:
diff --git a/src/sage/manifolds/differentiable/tensorfield_paral.py b/src/sage/manifolds/differentiable/tensorfield_paral.py
index b208f8b8825..312ec0fc26c 100644
--- a/src/sage/manifolds/differentiable/tensorfield_paral.py
+++ b/src/sage/manifolds/differentiable/tensorfield_paral.py
@@ -790,7 +790,7 @@ def add_comp(self, basis=None):
         for assignment.
 
         The components with respect to other frames on the same domain are
-        kept. To delete them them, use the method :meth:`set_comp` instead.
+        kept. To delete them, use the method :meth:`set_comp` instead.
 
         INPUT:
 
diff --git a/src/sage/manifolds/point.py b/src/sage/manifolds/point.py
index 66b9e309fb5..0f4a2201dac 100644
--- a/src/sage/manifolds/point.py
+++ b/src/sage/manifolds/point.py
@@ -74,6 +74,7 @@
 
 from sage.structure.element import Element
 from sage.misc.decorators import options
+from sage.symbolic.expression import Expression
 
 class ManifoldPoint(Element):
     r"""
@@ -610,9 +611,25 @@ def __eq__(self, other):
                 if chart in other._coordinates:
                     common_chart = chart
                     break
+        if common_chart is None:
+            # A commont chart is searched via a coordinate transformation,
+            # privileging the default chart
+            if def_chart in self._coordinates:
+                try:
+                    other.coordinates(def_chart)
+                    common_chart = def_chart
+                except ValueError:
+                    pass
+        if common_chart is None:
+            if def_chart in other._coordinates:
+                try:
+                    self.coordinates(def_chart)
+                    common_chart = def_chart
+                except ValueError:
+                    pass
         if common_chart is None:
             # At this stage, a commont chart is searched via a coordinate
-            # transformation:
+            # transformation from any chart
             for chart in self._coordinates:
                 try:
                     other.coordinates(chart)
@@ -634,7 +651,14 @@ def __eq__(self, other):
             #!# Another option would be:
             # raise ValueError("no common chart has been found to compare " +
             #                  "{} and {}".format(self, other))
-        return self._coordinates[common_chart] == other._coordinates[common_chart]
+        for xs, xo in zip(self._coordinates[common_chart],
+                          other._coordinates[common_chart]):
+            diff = xs - xo
+            if isinstance(diff, Expression) and not diff.is_trivial_zero():
+                return False
+            elif not (diff == 0):
+                return False
+        return True
 
     def __ne__(self, other):
         r"""
diff --git a/src/sage/manifolds/utilities.py b/src/sage/manifolds/utilities.py
index 5728fd31250..f3c4db83373 100644
--- a/src/sage/manifolds/utilities.py
+++ b/src/sage/manifolds/utilities.py
@@ -561,7 +561,7 @@ def _repr_(self):
 
         import re
 
-        # find all occurences of diff
+        # find all occurrences of diff
         list_d = []
         _list_derivatives(self, list_d)
 
@@ -584,7 +584,7 @@ def _repr_(self):
                 if bool(re.search(r'[+|-|/|*|^|(|)]', strv[i])):
                     strv[i] = "(" + strv[i] + ")"
 
-            # dictionary to group multiple occurences of differentiation: d/dxdx -> d/dx^2 etc.
+            # dictionary to group multiple occurrences of differentiation: d/dxdx -> d/dx^2 etc.
             occ = dict((i, strv[i] + "^" + str(diffargs.count(i))
                        if (diffargs.count(i)>1) else strv[i])
                        for i in diffargs)
@@ -654,7 +654,7 @@ def _latex_(self):
 
         import re
 
-        # find all occurences of diff
+        # find all occurrences of diff
         list_d = []
         _list_derivatives(self, list_d)
 
@@ -683,7 +683,7 @@ def _latex_(self):
                 if bool(re.search(r'[+|-|/|*|^|(|)]', val)):
                     latv[i] = "\left(" + latv[i] + "\\right)"
 
-            # dictionary to group multiple occurences of differentiation: d/dxdx -> d/dx^2 etc.
+            # dictionary to group multiple occurrences of differentiation: d/dxdx -> d/dx^2 etc.
             occ = {i: (latv[i] + "^" + latex(diffargs.count(i))
                        if diffargs.count(i) > 1 else latv[i])
                    for i in diffargs}
diff --git a/src/sage/matrix/constructor.pyx b/src/sage/matrix/constructor.pyx
index 577405a42fb..cf1194a7a60 100644
--- a/src/sage/matrix/constructor.pyx
+++ b/src/sage/matrix/constructor.pyx
@@ -12,6 +12,8 @@ General matrix Constructor
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
+
 import types
 from .matrix_space import MatrixSpace
 from sage.rings.ring import is_Ring
diff --git a/src/sage/matrix/matrix0.pyx b/src/sage/matrix/matrix0.pyx
index e48a7b2454b..f457fbdc48f 100644
--- a/src/sage/matrix/matrix0.pyx
+++ b/src/sage/matrix/matrix0.pyx
@@ -1680,8 +1680,6 @@ cdef class Matrix(sage.structure.element.Matrix):
             sage: A = random_matrix(ZZ, 100)
             sage: A.__repr__()
             '100 x 100 dense matrix over Integer Ring'
-            sage: print(A)
-            100 x 100 dense matrix over Integer Ring
 
         When a big matrix returned, include a hint on how to get the entries.
         This is a feature of the sage command-line::
@@ -1704,6 +1702,41 @@ cdef class Matrix(sage.structure.element.Matrix):
             s = 'dense'
         return "{} x {} {} matrix over {}".format(self._nrows, self._ncols, s, self.base_ring())
 
+    def __str__(self):
+        r"""
+        Return a string representation of this matrix. Unlike
+        ``__repr__`` (used by interactive sessions), this always prints
+        the matrix entries.
+
+        EXAMPLES::
+
+            sage: A = zero_matrix(ZZ, 20)
+            sage: A
+            20 x 20 dense matrix over Integer Ring (use the '.str()' method to see the entries)
+            sage: print(A)
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+            [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
+        """
+        return self.str()
+
     def str(self, rep_mapping=None, zero=None, plus_one=None, minus_one=None,
             *, unicode=False, shape=None):
         r"""
diff --git a/src/sage/matrix/matrix1.pyx b/src/sage/matrix/matrix1.pyx
index fe342e81010..4da7ecc306c 100644
--- a/src/sage/matrix/matrix1.pyx
+++ b/src/sage/matrix/matrix1.pyx
@@ -18,7 +18,7 @@ TESTS::
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-
+from __future__ import absolute_import
 
 from cpython cimport *
 
diff --git a/src/sage/matrix/matrix2.pyx b/src/sage/matrix/matrix2.pyx
index be6085a8bfd..f2f09095486 100644
--- a/src/sage/matrix/matrix2.pyx
+++ b/src/sage/matrix/matrix2.pyx
@@ -7239,7 +7239,7 @@ cdef class Matrix(Matrix1):
         INPUT:
 
         - ``check`` -- (default: ``False``) If ``True`` return a tuple of
-            the maximal matrix and the permutations taking taking ``self``
+            the maximal matrix and the permutations taking ``self``
             to the maximal matrix.
             If ``False``, return only the maximal matrix.
 
@@ -7347,7 +7347,7 @@ cdef class Matrix(Matrix1):
                     SN = [[k[1] for k in s] for s in SN]
                     # Maximal row in this entry of X = MS
                     nb = max(SN)
-                    # Number of occurences.
+                    # Number of occurrences.
                     n = [j for j in range(nrows - l) if SN[j] == nb]
                     # Now compare to our previous max
                     if b < nb:
@@ -10216,7 +10216,7 @@ cdef class Matrix(Matrix1):
         ALGORITHM:
 
         For each eigenvalue, this routine checks that the algebraic
-        multiplicity (number of occurences as a root of the characteristic
+        multiplicity (number of occurrences as a root of the characteristic
         polynomial) is equal to the geometric multiplicity (dimension
         of the eigenspace), which is sufficient to ensure a basis of
         eigenvectors for the columns of `S`.
diff --git a/src/sage/matrix/matrix_gfpn_dense.pyx b/src/sage/matrix/matrix_gfpn_dense.pyx
index 1a3ca2847c9..0d77633b89c 100644
--- a/src/sage/matrix/matrix_gfpn_dense.pyx
+++ b/src/sage/matrix/matrix_gfpn_dense.pyx
@@ -26,7 +26,7 @@ AUTHORS:
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from cysignals.memory cimport check_realloc, check_malloc, sig_free
 from cpython.bytes cimport PyBytes_AsString, PyBytes_FromStringAndSize
diff --git a/src/sage/matrix/matrix_integer_dense.pyx b/src/sage/matrix/matrix_integer_dense.pyx
index 783a170e051..d3db7138373 100644
--- a/src/sage/matrix/matrix_integer_dense.pyx
+++ b/src/sage/matrix/matrix_integer_dense.pyx
@@ -3271,7 +3271,7 @@ cdef class Matrix_integer_dense(Matrix_dense):
             sage: B.rational_reconstruction(389) == A/3
             True
 
-        TEST:
+        TESTS:
 
         Check that :trac:`9345` is fixed::
 
diff --git a/src/sage/matrix/matrix_integer_sparse.pyx b/src/sage/matrix/matrix_integer_sparse.pyx
index 2e2ce253dfb..2fdfe80caba 100644
--- a/src/sage/matrix/matrix_integer_sparse.pyx
+++ b/src/sage/matrix/matrix_integer_sparse.pyx
@@ -387,7 +387,7 @@ cdef class Matrix_integer_sparse(Matrix_sparse):
             [  7   2   2   3]
             [  4   3   4 5/7]
 
-        TEST:
+        TESTS:
 
         Check that :trac:`9345` is fixed::
 
diff --git a/src/sage/matrix/matrix_space.py b/src/sage/matrix/matrix_space.py
index deab170315d..4c37d6cf1bb 100644
--- a/src/sage/matrix/matrix_space.py
+++ b/src/sage/matrix/matrix_space.py
@@ -233,7 +233,7 @@ def __init__(self,  base_ring,
                         sparse=False,
                         implementation='flint'):
         """
-        TEST:
+        TESTS:
 
         We test that in the real or complex double dense case,
         conversion from the base ring is done by a call morphism.
diff --git a/src/sage/matrix/misc.pyx b/src/sage/matrix/misc.pyx
index 5eacdbe3a92..1c17c4595c3 100644
--- a/src/sage/matrix/misc.pyx
+++ b/src/sage/matrix/misc.pyx
@@ -67,7 +67,7 @@ def matrix_integer_dense_rational_reconstruction(Matrix_integer_dense A, Integer
         [ 7/3  2/3    6    1]
         [ 4/3    1  4/3  5/3]
 
-    TEST:
+    TESTS:
 
     Check that :trac:`9345` is fixed::
 
@@ -154,7 +154,7 @@ def matrix_integer_sparse_rational_reconstruction(Matrix_integer_sparse A, Integ
         [  7   2   2   3]
         [  4   3   4 5/7]
 
-    TEST:
+    TESTS:
 
     Check that :trac:`9345` is fixed::
 
diff --git a/src/sage/matrix/operation_table.py b/src/sage/matrix/operation_table.py
index 01b1ca70de1..e80a2ddf1d8 100644
--- a/src/sage/matrix/operation_table.py
+++ b/src/sage/matrix/operation_table.py
@@ -464,7 +464,7 @@ def _name_maker(self, names):
 
         EXAMPLES:
         This routine is tested extensively in the :class:`OperationTable`
-        and :meth:`change_names` methods.  So we just just demonstrate
+        and :meth:`change_names` methods.  So we just demonstrate
         the nature of the output here. ::
 
             sage: from sage.matrix.operation_table import OperationTable
diff --git a/src/sage/matrix/symplectic_basis.py b/src/sage/matrix/symplectic_basis.py
index 3230aeced4c..67013af915e 100644
--- a/src/sage/matrix/symplectic_basis.py
+++ b/src/sage/matrix/symplectic_basis.py
@@ -29,7 +29,7 @@
 Bourbaki, N.  Elements of Mathematics, Algebra III, Springer
 Verlag 2007.
 
-Kuperburg gives a more efficient and constructive exposition (see
+Kuperberg gives a more efficient and constructive exposition (see
 Theorem 18).
 
 Kuperberg, Greg.  Kasteleyn Cokernels.  Electr. J. Comb. 9(1), 2002.
diff --git a/src/sage/matroids/minor_matroid.py b/src/sage/matroids/minor_matroid.py
index 8fd9ba14165..c902612f198 100644
--- a/src/sage/matroids/minor_matroid.py
+++ b/src/sage/matroids/minor_matroid.py
@@ -83,10 +83,10 @@ class MinorMatroid(Matroid):
     r"""
     Minor of a matroid.
 
-    For some matroid representations it can be computationally expensive to
-    derive an explicit representation of a minor. This class wraps around any
-    matroid to provide an abstract minor. It also serves as default implemen-
-    tation.
+    For some matroid representations, it can be computationally
+    expensive to derive an explicit representation of a minor. This
+    class wraps around any matroid to provide an abstract minor. It
+    also serves as default implementation.
 
     Return a minor.
 
diff --git a/src/sage/misc/cachefunc.pyx b/src/sage/misc/cachefunc.pyx
index 2f70a5ba3de..df21e489a8b 100644
--- a/src/sage/misc/cachefunc.pyx
+++ b/src/sage/misc/cachefunc.pyx
@@ -468,7 +468,7 @@ the parent as its first argument::
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from cpython cimport PyObject
 
diff --git a/src/sage/misc/lazy_string.pyx b/src/sage/misc/lazy_string.pyx
index 66a0b5c89fa..44bd511844b 100644
--- a/src/sage/misc/lazy_string.pyx
+++ b/src/sage/misc/lazy_string.pyx
@@ -61,7 +61,7 @@ Note that the function is recomputed each time::
 #THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 #(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
 #OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from cpython.object cimport PyObject_Call, PyObject_RichCompare
 
diff --git a/src/sage/misc/misc.py b/src/sage/misc/misc.py
index 3cbb50e2a0f..14537f03953 100644
--- a/src/sage/misc/misc.py
+++ b/src/sage/misc/misc.py
@@ -140,7 +140,7 @@ def ECL_TMP():
         sage: ECL_TMP
         l'.../temp/.../ecl'
     """
-    d = os.path.join(SAGE_TMP, 'ecl')
+    d = os.path.join(str(SAGE_TMP), 'ecl')
     sage_makedirs(d)
     return d
 
@@ -154,7 +154,8 @@ def SPYX_TMP():
         sage: SPYX_TMP
         l'.../temp/.../spyx'
     """
-    return os.path.join(SAGE_TMP, 'spyx')
+    return os.path.join(str(SAGE_TMP), 'spyx')
+
 
 @lazy_string
 def SAGE_TMP_INTERFACE():
@@ -165,7 +166,7 @@ def SAGE_TMP_INTERFACE():
         sage: SAGE_TMP_INTERFACE
         l'.../temp/.../interface'
     """
-    d = os.path.join(SAGE_TMP, 'interface')
+    d = os.path.join(str(SAGE_TMP), 'interface')
     sage_makedirs(d)
     return d
 
diff --git a/src/sage/misc/parser.pyx b/src/sage/misc/parser.pyx
index bf2d2942b52..919d01ee630 100644
--- a/src/sage/misc/parser.pyx
+++ b/src/sage/misc/parser.pyx
@@ -19,6 +19,7 @@ AUTHOR:
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 from libc.string cimport strchr
 from cpython.bytes cimport PyBytes_FromStringAndSize
diff --git a/src/sage/misc/randstate.pyx b/src/sage/misc/randstate.pyx
index 266b03e35f2..79cc09b5936 100644
--- a/src/sage/misc/randstate.pyx
+++ b/src/sage/misc/randstate.pyx
@@ -407,6 +407,7 @@ Otherwise, it depends on what random number generator you want to use.
 Classes and methods
 ===================
 """
+from __future__ import absolute_import
 
 cdef extern from "stdlib.h":
     long c_libc_random "random"()
diff --git a/src/sage/misc/sageinspect.py b/src/sage/misc/sageinspect.py
index 0a3a9550f37..f9dffc1ce32 100644
--- a/src/sage/misc/sageinspect.py
+++ b/src/sage/misc/sageinspect.py
@@ -840,7 +840,7 @@ def _split_syntactical_unit(s):
         sage: _split_syntactical_unit('123')
         ('1', '23')
 
-    TEST:
+    TESTS:
 
     The following was fixed in :trac:`16309`::
 
@@ -1975,7 +1975,7 @@ def sage_getsourcelines(obj):
 
         sage: from sage.misc.sageinspect import sage_getsourcelines
         sage: sage_getsourcelines(matrix)[1]
-        26
+        28
         sage: sage_getsourcelines(matrix)[0][0][6:]
         'MatrixFactory(object):\n'
 
diff --git a/src/sage/misc/search.pyx b/src/sage/misc/search.pyx
index 92b48eaf688..fde47cf1174 100644
--- a/src/sage/misc/search.pyx
+++ b/src/sage/misc/search.pyx
@@ -22,6 +22,7 @@ extra comparison.   Also, the function names make more sense.
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 import bisect
 
diff --git a/src/sage/misc/stopgap.pyx b/src/sage/misc/stopgap.pyx
index 67e80a6853e..a8e605f5c88 100644
--- a/src/sage/misc/stopgap.pyx
+++ b/src/sage/misc/stopgap.pyx
@@ -11,6 +11,7 @@ Stopgaps
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 import warnings
 
diff --git a/src/sage/modular/abvar/abvar.py b/src/sage/modular/abvar/abvar.py
index 653e34ac0de..08cf14afc6f 100644
--- a/src/sage/modular/abvar/abvar.py
+++ b/src/sage/modular/abvar/abvar.py
@@ -4321,7 +4321,7 @@ def is_subvariety(self, other):
     def is_ambient(self):
         """
         Return True if this abelian variety attached to a modular symbols
-        space space is attached to the cuspidal subspace of the ambient
+        space is attached to the cuspidal subspace of the ambient
         modular symbols space.
 
         OUTPUT: bool
diff --git a/src/sage/modular/curves/notes/idea1.txt b/src/sage/modular/curves/notes/idea1.txt
deleted file mode 100644
index edcfc40f3b5..00000000000
--- a/src/sage/modular/curves/notes/idea1.txt
+++ /dev/null
@@ -1,37 +0,0 @@
-Extremely cool class idea! Very nice.
-
-A method which yields a smooth model (if known) of lowest degree
-would be nice too, if there was a natural one to choose from. For
-example, if the genus was one or 2 this might be known.
-
-+++++++++++++=
-
-William Stein wrote:
-> > David Joyner wrote:
->> >> BTW, I'm happy to document it properly if you want the
->> >> code. I guess you want this as a method in the Gamma0
->> >> class in congroup?
-> >
-> > I don't know where it should go.  Probably we should define
-> > a ModularCurve type and it should go there, right?
-> >
-> >     sage: X = ModularCurve(Gamma0(15))
-> >     sage: X.genus()
-> >     1
-> >     sage: X = X0(389)       # shorthand
-> >     sage: X.genus()
-> >     389
-> >
-> > And then there's a temptation to do more.
-> >     sage: X.canonical_embedding()
-> >     ...
-> >     sage: X.modular_polynomial()
-> >
-> >     sage: z = X(1 + 5*I)     # point defined by point in upper half
-> > plane.
-> >     sage: E = EllipticCurve("389A")
-> >     sage: phi = E.modular_parametrization(X)
-> >     sage: phi(z)             # point on E over C
-> >     sage: sum(...)           # construct a Heegner point.
-> >
-> > William
\ No newline at end of file
diff --git a/src/sage/modular/dirichlet.py b/src/sage/modular/dirichlet.py
index 99f427abde3..ca8d758cf09 100644
--- a/src/sage/modular/dirichlet.py
+++ b/src/sage/modular/dirichlet.py
@@ -1150,10 +1150,11 @@ def jacobi_sum(self, char, check=True):
 
         return sum([self(x) * char(1-x) for x in rings.IntegerModRing(self.modulus())])
 
-    def kloosterman_sum(self, a=1,b=0):
+    def kloosterman_sum(self, a=1, b=0):
         r"""
         Return the "twisted" Kloosterman sum associated to this Dirichlet character.
-        This includes Gauss sums, classical Kloosterman sums, Salie sums, etc.
+
+        This includes Gauss sums, classical Kloosterman sums, Salié sums, etc.
 
         The Kloosterman sum associated to `\chi` and the integers a,b is
 
@@ -1162,7 +1163,7 @@ def kloosterman_sum(self, a=1,b=0):
             K(a,b,\chi) = \sum_{r \in (\ZZ/m\ZZ)^\times} \chi(r)\,\zeta^{ar+br^{-1}},
 
         where `m` is the modulus of `\chi` and `\zeta` is a primitive
-        `m` th root of unity. This reduces to to the Gauss sum if `b=0`.
+        `m` th root of unity. This reduces to the Gauss sum if `b=0`.
 
         This method performs an exact calculation and returns an element of a
         suitable cyclotomic field; see also :meth:`.kloosterman_sum_numerical`,
@@ -1774,6 +1775,7 @@ def __setstate__(self, state):
         if element is not None:
             self.element.set_cache(element)
 
+
 class DirichletGroupFactory(UniqueFactory):
     r"""
     Construct a group of Dirichlet characters modulo `N`.
@@ -1984,7 +1986,8 @@ class DirichletGroupFactory(UniqueFactory):
         True
 
     """
-    def create_key(self, N, base_ring=None, zeta=None, zeta_order=None, names=None, integral=False):
+    def create_key(self, N, base_ring=None, zeta=None, zeta_order=None,
+                   names=None, integral=False):
         """
         Create a key that uniquely determines a Dirichlet group.
 
@@ -2034,8 +2037,14 @@ def create_key(self, N, base_ring=None, zeta=None, zeta_order=None, names=None,
             sage: G.order()
             4
 
+            sage: DirichletGroup(-33)
+            Traceback (most recent call last):
+            ...
+            ValueError: modulus should be positive            
         """
         modulus = rings.Integer(N)
+        if modulus <= 0:
+            raise ValueError('modulus should be positive')
 
         if base_ring is None:
             if not (zeta is None and zeta_order is None):
diff --git a/src/sage/modular/hecke/hecke_operator.py b/src/sage/modular/hecke/hecke_operator.py
index 2ee3fa76059..aacc9582a44 100644
--- a/src/sage/modular/hecke/hecke_operator.py
+++ b/src/sage/modular/hecke/hecke_operator.py
@@ -484,7 +484,7 @@ def _repr_(self):
             sage: ModularForms(Gamma0(100)).hecke_operator(4).matrix_form()._repr_()
             'Hecke operator on Modular Forms space of dimension 24 for Congruence Subgroup Gamma0(100) of weight 2 over Rational Field defined by:\n24 x 24 dense matrix over Rational Field'
         """
-        return "Hecke operator on %s defined by:\n%s"%(self.parent().module(), self.__matrix)
+        return "Hecke operator on %s defined by:\n%r" % (self.parent().module(), self.__matrix)
 
     def _latex_(self):
         r"""
diff --git a/src/sage/modular/hecke/morphism.py b/src/sage/modular/hecke/morphism.py
index b88d62dc680..b57536a9ed2 100644
--- a/src/sage/modular/hecke/morphism.py
+++ b/src/sage/modular/hecke/morphism.py
@@ -148,8 +148,8 @@ def _repr_(self):
         name = self.__name
         if name != '':
             name += ' '
-        return "Hecke module morphism %sdefined by the matrix\n%s\nDomain: %s\nCodomain: %s"%(\
-                name, str(self.matrix()), misc.strunc(self.domain()), misc.strunc(self.codomain()))
+        return "Hecke module morphism %sdefined by the matrix\n%r\nDomain: %s\nCodomain: %s"%(
+                name, self.matrix(), misc.strunc(self.domain()), misc.strunc(self.codomain()))
 
 # __mul__ method removed by David Loeffler 2009-04-14 as it is an exact duplicate of sage.modules.matrix_morphism.__mul__
 
diff --git a/src/sage/modular/modsym/tests.py b/src/sage/modular/modsym/tests.py
index 8b60a1e52e6..a4c7de3d313 100644
--- a/src/sage/modular/modsym/tests.py
+++ b/src/sage/modular/modsym/tests.py
@@ -122,7 +122,7 @@ def _modular_symbols_space(self):
 
     def _level_weight_sign(self):
         """
-        Return a triple containing a random choice of level from from
+        Return a triple containing a random choice of level from
         self.levels, weights from self.weights, and sign chosen
         randomly from [1, 0, -1].
 
diff --git a/src/sage/modular/pollack_stevens/dist.pxd b/src/sage/modular/pollack_stevens/dist.pxd
index 8eb4a3ff1e3..e437d4b0c84 100644
--- a/src/sage/modular/pollack_stevens/dist.pxd
+++ b/src/sage/modular/pollack_stevens/dist.pxd
@@ -5,7 +5,7 @@ from sage.rings.padics.pow_computer cimport PowComputer_class
 
 
 cdef class Dist(ModuleElement):
-    cpdef normalize(self)
+    cpdef normalize(self, include_zeroth_moment=*)
     cdef long ordp
     cpdef long _ord_p(self)
     cdef long _relprec(self)
@@ -15,6 +15,7 @@ cdef class Dist_vector(Dist):
     cdef public _moments
     cdef Dist_vector _new_c(self)
     cdef Dist_vector _addsub(self, Dist_vector right, bint negate)
+    cpdef _add_(self, other)
 
 # cdef class Dist_simple(Dist):
 #     cdef public _moments
diff --git a/src/sage/modular/pollack_stevens/dist.pyx b/src/sage/modular/pollack_stevens/dist.pyx
index b9e1df7854e..d348e4d1b4b 100644
--- a/src/sage/modular/pollack_stevens/dist.pyx
+++ b/src/sage/modular/pollack_stevens/dist.pyx
@@ -27,7 +27,8 @@ REFERENCES:
 #  the License, or (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
+
 from sage.structure.sage_object cimport SageObject
 from sage.structure.richcmp cimport richcmp_not_equal, rich_to_bool
 from sage.rings.integer_ring import ZZ
@@ -61,7 +62,7 @@ from sage.libs.flint.nmod_poly cimport (nmod_poly_init2_preinv,
 
 #from sage.libs.flint.ulong_extras cimport *
 
-from sigma0 import Sigma0
+from .sigma0 import Sigma0
 
 cdef long overflow = 1 << (4 * sizeof(long) - 1)
 cdef long underflow = -overflow
@@ -172,7 +173,7 @@ cdef class Dist(ModuleElement):
         """
         return self.parent().prime() ** (self.ordp) * self._moments
 
-    cpdef normalize(self):
+    cpdef normalize(self, include_zeroth_moment=True):
         r"""
         Normalize so that the precision of the `i`-th moment is `n-i`,
         where `n` is the number of moments.
@@ -1080,7 +1081,7 @@ cdef class Dist_vector(Dist):
         """
         return Integer(len(self._moments) + self.ordp)
 
-    cpdef normalize(self, include_zeroth_moment = True):
+    cpdef normalize(self, include_zeroth_moment=True):
         r"""
         Normalize by reducing modulo `Fil^N`, where `N` is the number of moments.
 
diff --git a/src/sage/modular/pollack_stevens/space.py b/src/sage/modular/pollack_stevens/space.py
index 0c8751ef0d8..64f9becee21 100644
--- a/src/sage/modular/pollack_stevens/space.py
+++ b/src/sage/modular/pollack_stevens/space.py
@@ -241,7 +241,7 @@ def __init__(self, group, coefficients, sign=0):
         """
         Module.__init__(self, coefficients.base_ring())
         if sign not in [0, -1, 1]:
-            # sign must be be 0, -1 or 1
+            # sign must be 0, -1 or 1
             raise ValueError("sign must be 0, -1, or 1")
         self._group = group
         self._coefficients = coefficients
diff --git a/src/sage/modules/free_module.py b/src/sage/modules/free_module.py
index b30690de769..f8d173b5fa1 100644
--- a/src/sage/modules/free_module.py
+++ b/src/sage/modules/free_module.py
@@ -1935,6 +1935,41 @@ def rank(self):
         """
         return self.__rank
 
+    def __bool__(self):
+        """
+        Return ``True`` if and only if the rank of this module is
+        non-zero. In other words, this returns ``False`` for the zero
+        module and ``True`` otherwise (apart from the exceptional case
+        where the base ring is the zero ring).
+
+        EXAMPLES::
+
+            sage: bool(QQ^0)
+            False
+            sage: bool(QQ^1)
+            True
+            sage: M = Matrix(2, 3, range(6))
+            sage: bool(M.right_kernel())
+            True
+            sage: bool(M.left_kernel())
+            False
+
+        When the base ring is the zero ring, we still look at the
+        "rank" (which may not be mathematically meaningful)::
+
+            sage: M = Integers(1)^4; M
+            Ambient free module of rank 4 over Ring of integers modulo 1
+            sage: M.rank()
+            4
+            sage: bool(M)
+            True
+            sage: M.cardinality()
+            1
+        """
+        return bool(self.rank())
+
+    __nonzero__ = __bool__
+
     def uses_ambient_inner_product(self):
         r"""
         Return ``True`` if the inner product on this module is
@@ -5492,11 +5527,11 @@ def _repr_(self):
         if self.is_sparse():
             s = "Sparse free module of degree %s and rank %s over %s\n"%(
                 self.degree(), self.rank(), self.base_ring()) + \
-                "User basis matrix:\n%s"%self.basis_matrix()
+                "User basis matrix:\n%r" % self.basis_matrix()
         else:
             s = "Free module of degree %s and rank %s over %s\n"%(
                 self.degree(), self.rank(), self.base_ring()) + \
-                "User basis matrix:\n%s"%self.basis_matrix()
+                "User basis matrix:\n%r" % self.basis_matrix()
         return s
 
     def _latex_(self):
@@ -6379,11 +6414,11 @@ def _repr_(self):
         if self.is_sparse():
             return "Sparse vector space of degree %s and dimension %s over %s\n"%(
                     self.degree(), self.dimension(), self.base_field()) + \
-                    "User basis matrix:\n%s"%self.basis_matrix()
+                    "User basis matrix:\n%r" % self.basis_matrix()
         else:
             return "Vector space of degree %s and dimension %s over %s\n"%(
                     self.degree(), self.dimension(), self.base_field()) + \
-                    "User basis matrix:\n%s"%self.basis_matrix()
+                    "User basis matrix:\n%r" % self.basis_matrix()
 
     def _denominator(self, B):
         """
@@ -6577,11 +6612,11 @@ def _repr_(self):
         if self.is_sparse():
             return "Sparse vector space of degree %s and dimension %s over %s\n"%(
                 self.degree(), self.dimension(), self.base_field()) + \
-                "Basis matrix:\n%s"%self.basis_matrix()
+                "Basis matrix:\n%r" % self.basis_matrix()
         else:
             return "Vector space of degree %s and dimension %s over %s\n"%(
                 self.degree(), self.dimension(), self.base_field()) + \
-                "Basis matrix:\n%s"%self.basis_matrix()
+                "Basis matrix:\n%r" % self.basis_matrix()
 
     def echelon_coordinates(self, v, check=True):
         """
diff --git a/src/sage/modules/free_module_morphism.py b/src/sage/modules/free_module_morphism.py
index 03f0c243a27..3a934ac2068 100644
--- a/src/sage/modules/free_module_morphism.py
+++ b/src/sage/modules/free_module_morphism.py
@@ -157,7 +157,7 @@ def _repr_(self):
             Domain: Vector space of dimension 40 over Rational Field
             Codomain: Vector space of dimension 40 over Rational Field
         """
-        r = "Free module morphism defined by the matrix\n{0}\nDomain: {1}\nCodomain: {2}"
+        r = "Free module morphism defined by the matrix\n{!r}\nDomain: {}\nCodomain: {}"
         return r.format(self.matrix(), self.domain(), self.codomain())
 
     def change_ring(self, R):
diff --git a/src/sage/modules/free_quadratic_module.py b/src/sage/modules/free_quadratic_module.py
index 815ad89d24c..ef0a2860201 100644
--- a/src/sage/modules/free_quadratic_module.py
+++ b/src/sage/modules/free_quadratic_module.py
@@ -1253,13 +1253,13 @@ def _repr_(self):
         if self.is_sparse():
             s = "Sparse free quadratic module of degree %s and rank %s over %s\n"%(
                 self.degree(), self.rank(), self.base_ring()) + \
-                "Basis matrix:\n%s\n" % self.basis_matrix() + \
-                "Inner product matrix:\n%s" % self.inner_product_matrix()
+                "Basis matrix:\n%r\n" % self.basis_matrix() + \
+                "Inner product matrix:\n%r" % self.inner_product_matrix()
         else:
             s = "Free quadratic module of degree %s and rank %s over %s\n"%(
                 self.degree(), self.rank(), self.base_ring()) + \
-                "Basis matrix:\n%s\n" % self.basis_matrix() + \
-                "Inner product matrix:\n%s" % self.inner_product_matrix()
+                "Basis matrix:\n%r\n" % self.basis_matrix() + \
+                "Inner product matrix:\n%r" % self.inner_product_matrix()
         return s
 
     def __cmp__(self, other):
@@ -1576,13 +1576,13 @@ def _repr_(self):
         if self.is_sparse():
             return "Sparse quadratic space of degree %s and dimension %s over %s\n"%(
                     self.degree(), self.dimension(), self.base_field()) + \
-                    "Basis matrix:\n%s"%self.basis_matrix() + \
-                    "Inner product matrix:\n%s" % self.inner_product_matrix()
+                    "Basis matrix:\n%r" % self.basis_matrix() + \
+                    "Inner product matrix:\n%r" % self.inner_product_matrix()
         else:
             return "Quadratic space of degree %s and dimension %s over %s\n"%(
                     self.degree(), self.dimension(), self.base_field()) + \
-                    "Basis matrix:\n%s\n"%self.basis_matrix() + \
-                    "Inner product matrix:\n%s" % self.inner_product_matrix()
+                    "Basis matrix:\n%r\n" % self.basis_matrix() + \
+                    "Inner product matrix:\n%r" % self.inner_product_matrix()
 
 class FreeQuadraticModule_submodule_field(
     free_module.FreeModule_submodule_field, FreeQuadraticModule_submodule_with_basis_field):
@@ -1685,13 +1685,13 @@ def _repr_(self):
         if self.is_sparse():
             return "Sparse quadratic space of degree %s and dimension %s over %s\n"%(
                 self.degree(), self.dimension(), self.base_field()) + \
-                "Basis matrix:\n%s\n" % self.basis_matrix() + \
-                "Inner product matrix:\n%s" % self.inner_product_matrix()
+                "Basis matrix:\n%r\n" % self.basis_matrix() + \
+                "Inner product matrix:\n%r" % self.inner_product_matrix()
         else:
             return "Quadratic space of degree %s and dimension %s over %s\n"%(
                 self.degree(), self.dimension(), self.base_field()) + \
-                "Basis matrix:\n%s\n" % self.basis_matrix() + \
-                "Inner product matrix:\n%s" % self.inner_product_matrix()
+                "Basis matrix:\n%r\n" % self.basis_matrix() + \
+                "Inner product matrix:\n%r" % self.inner_product_matrix()
 
 #class RealDoubleQuadraticSpace_class(free_module.RealDoubleVectorSpace_class, FreeQuadraticModule_ambient_field):
 #    def __init__(self, dimension, inner_product_matrix, sparse=False):
diff --git a/src/sage/modules/vector_space_morphism.py b/src/sage/modules/vector_space_morphism.py
index 22456847516..000a7c80f5e 100644
--- a/src/sage/modules/vector_space_morphism.py
+++ b/src/sage/modules/vector_space_morphism.py
@@ -954,7 +954,7 @@ def _repr_(self):
         """
         m = self.matrix()
         msg = ("Vector space morphism represented by the matrix:\n",
-               "{0}\n",
-               "Domain: {1}\n",
-               "Codomain: {2}")
+               "{!r}\n",
+               "Domain: {}\n",
+               "Codomain: {}")
         return ''.join(msg).format(m, self.domain(), self.codomain())
diff --git a/src/sage/numerical/backends/coin_backend.pxd b/src/sage/numerical/backends/coin_backend.pxd
index c45427cb2e5..0a24a24ad94 100644
--- a/src/sage/numerical/backends/coin_backend.pxd
+++ b/src/sage/numerical/backends/coin_backend.pxd
@@ -129,7 +129,7 @@ cdef extern from "coin/OsiSolverInterface.hpp":
         void getBInvARow(int row, double* z, double * slack)
         void getBInvACol(int col, double* vec)
 
-        # Get indicies of basic variables
+        # Get indices of basic variables
         void getBasics(int* index)
 
         # Get objective coefficients
diff --git a/src/sage/numerical/backends/cplex_backend.pyx b/src/sage/numerical/backends/cplex_backend.pyx
index ca7cb7d745f..d134d5b6214 100644
--- a/src/sage/numerical/backends/cplex_backend.pyx
+++ b/src/sage/numerical/backends/cplex_backend.pyx
@@ -1535,7 +1535,7 @@ cdef class CPLEXBackend(GenericBackend):
             sage: p.solver_parameter("logfile")            # optional - CPLEX
             ''
 
-        TEST:
+        TESTS:
 
         Print the logfile's content (through :class:`MixedIntegerLinearProgram`)::
 
diff --git a/src/sage/numerical/backends/generic_sdp_backend.pyx b/src/sage/numerical/backends/generic_sdp_backend.pyx
index 90b0fae9caa..5aed5aa6cd3 100644
--- a/src/sage/numerical/backends/generic_sdp_backend.pyx
+++ b/src/sage/numerical/backends/generic_sdp_backend.pyx
@@ -6,7 +6,7 @@ interface with a SDP Solver. All these methods immediately raise
 ``NotImplementedError`` exceptions when called, and are obviously
 meant to be replaced by the solver-specific method. This file can also
 be used as a template to create a new interface : one would only need
-to replace the occurences of ``"Nonexistent_SDP_solver"`` by the
+to replace the occurrences of ``"Nonexistent_SDP_solver"`` by the
 solver's name, and replace ``GenericSDPBackend`` by
 ``SolverName(GenericSDPBackend)`` so that the new solver extends this
 class.
diff --git a/src/sage/numerical/linear_functions.pxd b/src/sage/numerical/linear_functions.pxd
index bf447586b6e..284a5b34aa7 100644
--- a/src/sage/numerical/linear_functions.pxd
+++ b/src/sage/numerical/linear_functions.pxd
@@ -13,6 +13,7 @@ cdef class LinearFunctionsParent_class(Parent):
 
 cdef class LinearFunction(LinearFunctionOrConstraint):
     cdef dict _f
+    cpdef _add_(self, other)
     cpdef iteritems(self)
     cpdef _acted_upon_(self, x, bint self_on_left)
     cpdef is_zero(self)
diff --git a/src/sage/numerical/linear_tensor_element.pxd b/src/sage/numerical/linear_tensor_element.pxd
index 5de33004079..fe4fb18fb88 100644
--- a/src/sage/numerical/linear_tensor_element.pxd
+++ b/src/sage/numerical/linear_tensor_element.pxd
@@ -2,3 +2,4 @@ from sage.structure.element cimport Element, ModuleElement
 
 cdef class LinearTensor(ModuleElement):
     cpdef dict _f
+    cpdef _add_(self, other)
diff --git a/src/sage/numerical/mip.pyx b/src/sage/numerical/mip.pyx
index e2417db01cd..7376141f2bd 100644
--- a/src/sage/numerical/mip.pyx
+++ b/src/sage/numerical/mip.pyx
@@ -611,7 +611,7 @@ cdef class MixedIntegerLinearProgram(SageObject):
             sage: cp.solve()
             6.0
 
-        TEST:
+        TESTS:
 
         Test that `deepcopy` makes actual copies but preserves identities::
 
diff --git a/src/sage/plot/graphics.py b/src/sage/plot/graphics.py
index 2c8a48c1085..8fd3998927b 100644
--- a/src/sage/plot/graphics.py
+++ b/src/sage/plot/graphics.py
@@ -693,8 +693,7 @@ def axes_labels(self, l=None):
             sage: c = circle((0,0), 1)
             sage: c.axes_labels(['axe des abscisses', u'axe des ordonnées'])
             sage: c._axes_labels
-            ('axe des abscisses', u'axe des ordonn\xc3\xa9es')
-
+            ('axe des abscisses', u'axe des ordonn\xe9es')
         """
         if l is None:
             try:
diff --git a/src/sage/plot/plot3d/implicit_surface.pyx b/src/sage/plot/plot3d/implicit_surface.pyx
index faa67a0b13c..ff71df5936b 100644
--- a/src/sage/plot/plot3d/implicit_surface.pyx
+++ b/src/sage/plot/plot3d/implicit_surface.pyx
@@ -49,7 +49,6 @@ AUTHORS:
 #  *  Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
 #*****************************************************************************
 
-
 # There's a framework in here for computing multiple isosurfaces of a
 # single function.  Currently, it's only used for a single
 # implicit_plot3d where contour=... is given a list, but it's ready to
@@ -77,6 +76,7 @@ AUTHORS:
 # n^3 memory when it reads the JVXL file, but that might be on a different
 # computer; Tachyon would only allocate memory proportional to the
 # output size.)
+from __future__ import absolute_import
 
 cimport numpy as np
 import numpy as np
diff --git a/src/sage/plot/plot3d/introduction.py b/src/sage/plot/plot3d/introduction.py
index 15d78e85d82..6789a52fd43 100644
--- a/src/sage/plot/plot3d/introduction.py
+++ b/src/sage/plot/plot3d/introduction.py
@@ -50,7 +50,7 @@
 
 -  :func:`~sage.plot.plot3d.shapes.ColorCube` - a cube with given size and sides with given colors
 
--  :func:`~sage.plot.plot3d.shapes.LineSegment` - a line segment, which is drawn as a cylinder from start to end with radius radius
+-  :func:`~sage.plot.plot3d.shapes.LineSegment` - a line segment, which is drawn as a cylinder from start to end with given radius
 
 -  :func:`~sage.plot.plot3d.shapes2.line3d` - a 3d line joining a sequence of points
 
diff --git a/src/sage/plot/plot3d/shapes.pyx b/src/sage/plot/plot3d/shapes.pyx
index 0dbe6fbf1ee..4726c5aca9b 100644
--- a/src/sage/plot/plot3d/shapes.pyx
+++ b/src/sage/plot/plot3d/shapes.pyx
@@ -599,7 +599,7 @@ draw %s width %s {%s %s %s} {%s %s %s}\n%s
 def LineSegment(start, end, thickness=1, radius=None, **kwds):
     """
     Create a line segment, which is drawn as a cylinder from start to
-    end with radius radius.
+    end with radius ``radius``.
 
     EXAMPLES::
 
diff --git a/src/sage/plot/streamline_plot.py b/src/sage/plot/streamline_plot.py
index 8ea73fa5151..f3da57c3706 100644
--- a/src/sage/plot/streamline_plot.py
+++ b/src/sage/plot/streamline_plot.py
@@ -153,7 +153,7 @@ def streamline_plot(f_g, xrange, yrange, **options):
     two variables `x` and `y`.
 
     If given two functions `(f(x,y), g(x,y))`, then this function plots
-    steamlines in the vector field over the specified ranges with ``xrange``
+    streamlines in the vector field over the specified ranges with ``xrange``
     being of `x`, denoted by ``xvar`` below, between ``xmin`` and ``xmax``,
     and ``yrange`` similarly (see below). ::
 
diff --git a/src/sage/quivers/algebra_elements.pxd b/src/sage/quivers/algebra_elements.pxd
index dd49e175449..61aa11c79a8 100644
--- a/src/sage/quivers/algebra_elements.pxd
+++ b/src/sage/quivers/algebra_elements.pxd
@@ -83,6 +83,8 @@ cdef class PathAlgebraElement(RingElement):
     # functions.
     cdef path_order_t cmp_terms
     cdef long _hash
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cpdef ssize_t degree(self) except -2
     cpdef dict monomial_coefficients(self)
     cpdef list coefficients(self)
diff --git a/src/sage/quivers/homspace.py b/src/sage/quivers/homspace.py
index 02937e7fb32..d44d2ccf6d2 100644
--- a/src/sage/quivers/homspace.py
+++ b/src/sage/quivers/homspace.py
@@ -94,7 +94,7 @@ def __init__(self, domain, codomain, category=None):
         # and N to the vertices of the quiver.  Each coordinate represents a
         # single entry in one of those matrices.
 
-        # Get the quiver and base ring and check they they are the same for
+        # Get the quiver and base ring and check that they are the same for
         # both modules
         if domain._semigroup != codomain._semigroup:
             raise ValueError("representations are not over the same quiver")
diff --git a/src/sage/quivers/paths.pxd b/src/sage/quivers/paths.pxd
index 09fcddc380a..79f7598f913 100644
--- a/src/sage/quivers/paths.pxd
+++ b/src/sage/quivers/paths.pxd
@@ -5,6 +5,8 @@ cdef class QuiverPath(MonoidElement):
     cdef biseq_t _path
     cdef int _start, _end
     cdef QuiverPath _new_(self, int start, int end)
+    cpdef _mul_(self, other)
+    cpdef _mod_(self, right)
     cpdef tuple complement(self, QuiverPath subpath)
     cpdef bint has_subpath(self, QuiverPath subpath) except -1
     cpdef bint has_prefix(self, QuiverPath subpath) except -1
diff --git a/src/sage/repl/display/fancy_repr.py b/src/sage/repl/display/fancy_repr.py
index 8ffc817b31f..a2df28fcb30 100644
--- a/src/sage/repl/display/fancy_repr.py
+++ b/src/sage/repl/display/fancy_repr.py
@@ -209,7 +209,7 @@ def __call__(self, obj, p, cycle):
         if obj.nrows() < max_rows and obj.ncols() < max_cols:
             return False
         p.text(
-            str(obj) + " (use the '.str()' method to see the entries)"
+            repr(obj) + " (use the '.str()' method to see the entries)"
         )
         return True
 
diff --git a/src/sage/repl/rich_output/backend_ipython.py b/src/sage/repl/rich_output/backend_ipython.py
index 894b7d274e8..5085c3dca58 100644
--- a/src/sage/repl/rich_output/backend_ipython.py
+++ b/src/sage/repl/rich_output/backend_ipython.py
@@ -236,7 +236,7 @@ def displayhook(self, plain_text, rich_output):
             sage: from sage.repl.rich_output import get_display_manager
             sage: dm = get_display_manager()
             sage: dm.displayhook(Foo())
-            ({u'text/plain': u'Mot\xc3\xb6rhead'}, {})
+            ({u'text/plain': u'Mot\xf6rhead'}, {})
         """
         if isinstance(rich_output, OutputPlainText):
             return ({u'text/plain': rich_output.text.get_unicode()}, {})
diff --git a/src/sage/repl/rich_output/display_manager.py b/src/sage/repl/rich_output/display_manager.py
index ab032177010..fbb38f8a8ee 100644
--- a/src/sage/repl/rich_output/display_manager.py
+++ b/src/sage/repl/rich_output/display_manager.py
@@ -107,7 +107,7 @@ def __init__(self, display_manager, output_classes):
 
         In the context, the output is restricted to the output
         container types listed in ``output_classes``. Additionally,
-        display preferences are changed not not show graphics.
+        display preferences are changed not to show graphics.
 
         INPUT:
 
diff --git a/src/sage/rings/algebraic_closure_finite_field.py b/src/sage/rings/algebraic_closure_finite_field.py
index 86fa5e8c37e..864181ec867 100644
--- a/src/sage/rings/algebraic_closure_finite_field.py
+++ b/src/sage/rings/algebraic_closure_finite_field.py
@@ -63,6 +63,8 @@
 from sage.rings.finite_rings.finite_field_base import is_FiniteField
 from sage.rings.ring import Field
 from sage.structure.element import FieldElement
+from sage.structure.richcmp import richcmp
+
 
 class AlgebraicClosureFiniteFieldElement(FieldElement):
     """
@@ -164,7 +166,7 @@ def _repr_(self):
         """
         return self._value._repr_()
 
-    def __cmp__(self, right):
+    def _richcmp_(self, right, op):
         """
         Compare ``self`` with ``right``.
 
@@ -173,10 +175,9 @@ def __cmp__(self, right):
             sage: F = GF(3).algebraic_closure()
             sage: F.gen(2) == F.gen(3)
             False
-
         """
         x, y = self.parent()._to_common_subfield(self, right)
-        return cmp(x, y)
+        return richcmp(x, y, op)
 
     def __pow__(self, exp):
         r"""
@@ -558,7 +559,7 @@ def __init__(self, base_ring, name, category=None):
         Field.__init__(self, base_ring=base_ring, names=name,
                        normalize=False, category=category)
 
-    def __cmp__(self, other):
+    def __eq__(self, other):
         """
         Compare ``self`` with ``other``.
 
@@ -570,15 +571,28 @@ def __cmp__(self, other):
             sage: F5 = GF(5).algebraic_closure()
             sage: F3 == F5
             False
-
         """
         if self is other:
-            return 0
-        c = cmp(type(self), type(other))
-        if c != 0:
-            return c
-        return cmp((self.base_ring(), self.variable_name(), self.category()),
-                   (other.base_ring(), other.variable_name(), other.category()))
+            return True
+        if type(self) != type(other):
+            return False
+        return ((self.base_ring(), self.variable_name(), self.category()) ==
+                (other.base_ring(), other.variable_name(), other.category()))
+
+    def __ne__(self, other):
+        """
+        Check whether ``self`` and ``other`` are not equal.
+
+        TESTS::
+
+            sage: F3 = GF(3).algebraic_closure()
+            sage: F3 != F3
+            False
+            sage: F5 = GF(5).algebraic_closure()
+            sage: F3 != F5
+            True
+        """
+        return not (self == other)
 
     def cardinality(self):
         """
@@ -1029,7 +1043,8 @@ def _factor_univariate_polynomial(self, p, **kwds):
         R = p.parent()
         return Factorization([(R([-root, self.one()]), m) for root, m in p.roots()], unit=p[p.degree()])
 
-class AlgebraicClosureFiniteField_pseudo_conway(AlgebraicClosureFiniteField_generic, WithEqualityById):
+
+class AlgebraicClosureFiniteField_pseudo_conway(WithEqualityById, AlgebraicClosureFiniteField_generic):
     """
     Algebraic closure of a finite field, constructed using
     pseudo-Conway polynomials.
diff --git a/src/sage/rings/asymptotic/asymptotic_ring.py b/src/sage/rings/asymptotic/asymptotic_ring.py
index 8ac348e5438..1703babd693 100644
--- a/src/sage/rings/asymptotic/asymptotic_ring.py
+++ b/src/sage/rings/asymptotic/asymptotic_ring.py
@@ -1075,7 +1075,7 @@ def monomial_coefficient(self, monomial):
         INPUT:
 
         - ``monomial`` -- a monomial element which can be converted
-          into the the asymptotic ring of this element
+          into the asymptotic ring of this element
 
         OUTPUT:
 
diff --git a/src/sage/rings/cfinite_sequence.py b/src/sage/rings/cfinite_sequence.py
index 5df326bcd4f..468a540618e 100644
--- a/src/sage/rings/cfinite_sequence.py
+++ b/src/sage/rings/cfinite_sequence.py
@@ -338,7 +338,7 @@ def __init__(self, parent, ogf):
         INPUT:
 
         - ``ogf`` -- the ordinary generating function, a fraction of polynomials over the rationals
-        - ``parent`` --  the parent of the C-Finite sequence, an occurence of :class:`CFiniteSequences`
+        - ``parent`` --  the parent of the C-Finite sequence, an occurrence of :class:`CFiniteSequences`
 
         OUTPUT:
 
diff --git a/src/sage/rings/complex_arb.pxd b/src/sage/rings/complex_arb.pxd
index 0f9882f36ef..8c2e5152807 100644
--- a/src/sage/rings/complex_arb.pxd
+++ b/src/sage/rings/complex_arb.pxd
@@ -14,6 +14,8 @@ cdef int acb_to_ComplexIntervalFieldElement(
 cdef class ComplexBall(RingElement):
     cdef acb_t value
     cdef ComplexBall _new(self)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cpdef ComplexIntervalFieldElement _complex_mpfi_(self, parent)
     cpdef RealBall real(self)
     cpdef RealBall imag(self)
diff --git a/src/sage/rings/complex_arb.pyx b/src/sage/rings/complex_arb.pyx
index 775c1b4ba9c..dfa7be2b651 100644
--- a/src/sage/rings/complex_arb.pyx
+++ b/src/sage/rings/complex_arb.pyx
@@ -128,6 +128,7 @@ Classes and Methods
 #  the License, or (at your option) any later version.
 #                http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 import operator
 from cysignals.signals cimport sig_on, sig_str, sig_off, sig_error
diff --git a/src/sage/rings/complex_double.pxd b/src/sage/rings/complex_double.pxd
index 012320f8cda..929a509c5c8 100644
--- a/src/sage/rings/complex_double.pxd
+++ b/src/sage/rings/complex_double.pxd
@@ -13,5 +13,7 @@ cdef class ComplexDoubleField_class(sage.rings.ring.Field):
 cdef class ComplexDoubleElement(sage.structure.element.FieldElement):
     cdef gsl_complex _complex
     cdef ComplexDoubleElement _new_c(self, gsl_complex x)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
 
 cdef ComplexDoubleElement new_ComplexDoubleElement()
diff --git a/src/sage/rings/complex_interval.pxd b/src/sage/rings/complex_interval.pxd
index 6734dbf3bbf..50412f4be2a 100644
--- a/src/sage/rings/complex_interval.pxd
+++ b/src/sage/rings/complex_interval.pxd
@@ -8,6 +8,8 @@ cdef class ComplexIntervalFieldElement(sage.structure.element.FieldElement):
     cdef mpfi_t __im
     cdef mp_prec_t _prec
 
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cdef RealIntervalFieldElement abs_c(ComplexIntervalFieldElement self)
     cdef RealIntervalFieldElement norm_c(ComplexIntervalFieldElement self)
 
diff --git a/src/sage/rings/complex_mpc.pxd b/src/sage/rings/complex_mpc.pxd
index 48c03e39627..11e1c162b59 100644
--- a/src/sage/rings/complex_mpc.pxd
+++ b/src/sage/rings/complex_mpc.pxd
@@ -7,6 +7,8 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
     cdef mpc_t value
     cdef char init
     cdef MPComplexNumber _new(self)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
 
 cdef class MPComplexField_class(sage.rings.ring.Field):
     cdef readonly int __prec
diff --git a/src/sage/rings/complex_mpc.pyx b/src/sage/rings/complex_mpc.pyx
index f08b543c04c..8cad7fbec0f 100644
--- a/src/sage/rings/complex_mpc.pyx
+++ b/src/sage/rings/complex_mpc.pyx
@@ -916,7 +916,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
             sage: MPComplexField()(2, -3) # indirect doctest
             2.00000000000000 - 3.00000000000000*I
         """
-        return self.str()
+        return self.str(truncate=True)
 
     def _latex_(self):
         """
@@ -928,7 +928,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
             2.00000000000000 - 3.00000000000000i
         """
         import re
-        s = self.str().replace('*I', 'i')
+        s = repr(self).replace('*I', 'i')
         return re.sub(r"e(-?\d+)", r" \\times 10^{\1}", s)
 
     def __hash__(self):
@@ -1030,40 +1030,40 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
         mpfr_set (y.value, self.value.im, (<RealField_class>y._parent).rnd)
         return y
 
-    def str(self, int base=10, int truncate=True):
+    def str(self, base=10, **kwds):
         """
         Return a string of ``self``.
 
         INPUT:
 
-        - ``base`` -- base for output
+        - ``base`` -- (default: 10) base for output
 
-        - ``truncate`` -- if ``True``, round off the last digits in printing
-          to lessen confusing base-2 roundoff issues.
+        - ``**kwds`` -- other arguments to pass to the ``str()``
+          method of the real numbers in the real and imaginary parts.
 
         EXAMPLES::
 
             sage: MPC = MPComplexField(64)
             sage: z = MPC(-4, 3)/7
             sage: z.str()
-            '-0.571428571428571429 + 0.428571428571428571*I'
+            '-0.571428571428571428564 + 0.428571428571428571436*I'
             sage: z.str(16)
             '-0.92492492492492490 + 0.6db6db6db6db6db70*I'
             sage: z.str(truncate=True)
             '-0.571428571428571429 + 0.428571428571428571*I'
-            sage: z.str(2, True)
+            sage: z.str(2)
             '-0.1001001001001001001001001001001001001001001001001001001001001001 + 0.01101101101101101101101101101101101101101101101101101101101101110*I'
         """
         s = ""
         if self.real() != 0:
-            s = self.real().str(base, truncate=truncate)
+            s = self.real().str(base, **kwds)
         if self.imag() != 0:
             if mpfr_signbit(self.value.im):
-                s += ' - ' + (-self.imag()).str(base, truncate=truncate) + '*I'
+                s += ' - ' + (-self.imag()).str(base, **kwds) + '*I'
             else:
                 if s:
                     s += ' + '
-                s += self.imag().str(base, truncate=truncate) + '*I'
+                s += self.imag().str(base, **kwds) + '*I'
         if not s:
             return "0"
         return s
diff --git a/src/sage/rings/complex_number.pxd b/src/sage/rings/complex_number.pxd
index beedab2558d..a3f12ef36fe 100644
--- a/src/sage/rings/complex_number.pxd
+++ b/src/sage/rings/complex_number.pxd
@@ -9,6 +9,8 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
     cdef object _multiplicative_order
     cdef int _prec
 
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cdef RealNumber abs_c(ComplexNumber self)
     cdef RealNumber norm_c(ComplexNumber self)
 
diff --git a/src/sage/rings/complex_number.pyx b/src/sage/rings/complex_number.pyx
index 3ed2265616c..133431803bc 100644
--- a/src/sage/rings/complex_number.pyx
+++ b/src/sage/rings/complex_number.pyx
@@ -220,7 +220,7 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
             sage: s1 == CC(gp(s1))
             True
         """
-        return self.str(truncate=False)
+        return self.str()
 
     def _maxima_init_(self, I=None):
         """
@@ -234,7 +234,7 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
             sage: CC(.5 + I)._maxima_init_()
             '0.50000000000000000 + 1.0000000000000000*%i'
         """
-        return self.str(truncate=False, istr='%i')
+        return self.str(istr='%i')
 
     @property
     def __array_interface__(self):
@@ -366,7 +366,7 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
             sage: a._repr_()
             '2.00000000000000 + 1.00000000000000*I'
         """
-        return self.str(10)
+        return self.str(truncate=True)
 
     def __hash__(self):
         """
@@ -459,25 +459,27 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         """
         self._multiplicative_order = integer.Integer(n)
 
-    def str(self, base=10, truncate=True, istr='I'):
+    def str(self, base=10, istr='I', **kwds):
         r"""
         Return a string representation of ``self``.
 
         INPUT:
 
-        - ``base`` --  (Default: 10) The base to use for printing
+        - ``base`` -- (default: 10) base for output
 
-        - ``truncate`` -- (Default: ``True``) Whether to print fewer
-          digits than are available, to mask errors in the last bits.
+        - ``istr`` -- (default: ``I``) String representation of the complex unit
 
-        - ``istr`` -- (Default: ``I``) String representation of the complex unit
+        - ``**kwds`` -- other arguments to pass to the ``str()``
+          method of the real numbers in the real and imaginary parts.
 
         EXAMPLES::
 
             sage: a = CC(pi + I*e)
-            sage: a.str()
+            sage: a
+            3.14159265358979 + 2.71828182845905*I
+            sage: a.str(truncate=True)
             '3.14159265358979 + 2.71828182845905*I'
-            sage: a.str(truncate=False)
+            sage: a.str()
             '3.1415926535897931 + 2.7182818284590451*I'
             sage: a.str(base=2)
             '11.001001000011111101101010100010001000010110100011000 + 10.101101111110000101010001011000101000101011101101001*I'
@@ -490,23 +492,23 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
             sage: CC(0)
             0.000000000000000
             sage: CC.0.str(istr='%i')
-            '1.00000000000000*%i'
+            '1.0000000000000000*%i'
         """
         s = ""
-        if self.real() != 0:
-            s = self.real().str(base, truncate=truncate)
-        if self.imag() != 0:
-            y  =  self.imag()
-            if s!="":
+        if self.real():
+            s = self.real().str(base, **kwds)
+        if self.imag():
+            y = self.imag()
+            if s:
                 if y < 0:
-                    s = s+" - "
+                    s += " - "
                     y = -y
                 else:
-                    s = s+" + "
-            s = s+"{ystr}*{istr}".format(ystr=y.str(base, truncate=truncate),
-                    istr=istr)
-        if len(s) == 0:
-            s = self.real().str(base, truncate=truncate)
+                    s += " + "
+            ystr = y.str(base, **kwds)
+            s += ystr + "*" + istr
+        if not s:
+            s = self.real().str(base, **kwds)
         return s
 
     def _latex_(self):
@@ -540,7 +542,7 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
             '-\\infty'
         """
         import re
-        s = self.str().replace('*I', 'i').replace('infinity','\\infty')
+        s = repr(self).replace('*I', 'i').replace('infinity','\\infty')
         return re.sub(r"e(-?\d+)", r" \\times 10^{\1}", s)
 
     def __pari__(self):
@@ -854,8 +856,8 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
             sage: magma(ComplexField(200)(1/3)) # indirect, optional - magma
             0.333333333333333333333333333333333333333333333333333333333333
         """
-        real_string = self.real().str(truncate=False)
-        imag_string = self.imag().str(truncate=False)
+        real_string = self.real().str()
+        imag_string = self.imag().str()
         digit_precision_bound = len(real_string)
         return "%s![%sp%s, %sp%s]" % (self.parent()._magma_init_(magma),
                                       real_string, digit_precision_bound,
diff --git a/src/sage/rings/finite_rings/element_givaro.pyx b/src/sage/rings/finite_rings/element_givaro.pyx
index c9dc9a1dac0..422b948b3aa 100644
--- a/src/sage/rings/finite_rings/element_givaro.pyx
+++ b/src/sage/rings/finite_rings/element_givaro.pyx
@@ -1199,7 +1199,7 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
             sage: ~a*a
             1
 
-        TEST:
+        TESTS:
 
         Check that trying to invert zero raises an error
         (see :trac:`12217`)::
diff --git a/src/sage/rings/finite_rings/integer_mod.pyx b/src/sage/rings/finite_rings/integer_mod.pyx
index f3441cc61fd..18a393aa73f 100644
--- a/src/sage/rings/finite_rings/integer_mod.pyx
+++ b/src/sage/rings/finite_rings/integer_mod.pyx
@@ -67,7 +67,7 @@ TESTS::
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function, division
+from __future__ import print_function, division, absolute_import
 
 from cysignals.signals cimport sig_on, sig_off
 
@@ -143,8 +143,8 @@ def Mod(n, m, parent=None):
 
     # m is non-zero, so return n mod m
     cdef IntegerMod_abstract x
-    import integer_mod_ring
-    x = IntegerMod(integer_mod_ring.IntegerModRing(m), n)
+    from .integer_mod_ring import IntegerModRing
+    x = IntegerMod(IntegerModRing(m), n)
     if parent is None:
         return x
     x._parent = parent
@@ -388,7 +388,7 @@ cdef class IntegerMod_abstract(FiniteRingElement):
         if not isinstance(self, IntegerMod_abstract):
             # something % Mod(x,y) makes no sense
             return NotImplemented
-        from integer_mod_ring import IntegerModRing
+        from .integer_mod_ring import IntegerModRing
         R = IntegerModRing(modulus)
         if (<Element>self)._parent._IntegerModRing_generic__order % R.order():
             raise ArithmeticError(f"reduction modulo {modulus!r} not defined")
@@ -1933,8 +1933,8 @@ cdef class IntegerMod_gmp(IntegerMod_abstract):
 
         modulus = self.__modulus.sageInteger
         other_modulus = other.__modulus.sageInteger
-        import integer_mod_ring
-        lift = IntegerMod_gmp(integer_mod_ring.IntegerModRing(modulus*other_modulus), None, empty=True)
+        from .integer_mod_ring import IntegerModRing
+        lift = IntegerMod_gmp(IntegerModRing(modulus*other_modulus), None, empty=True)
         try:
             if mpz_cmp(self.value, other.value) > 0:
                 x = (other - IntegerMod_gmp(other._parent, self.lift())) / IntegerMod_gmp(other._parent, modulus)
@@ -2350,8 +2350,8 @@ cdef class IntegerMod_int(IntegerMod_abstract):
         cdef IntegerMod_int lift
         cdef int_fast32_t x
 
-        import integer_mod_ring
-        lift = IntegerMod_int(integer_mod_ring.IntegerModRing(self.__modulus.int32 * other.__modulus.int32), None, empty=True)
+        from .integer_mod_ring import IntegerModRing
+        lift = IntegerMod_int(IntegerModRing(self.__modulus.int32 * other.__modulus.int32), None, empty=True)
 
         try:
             x = (other.ivalue - self.ivalue % other.__modulus.int32) * mod_inverse_int(self.__modulus.int32, other.__modulus.int32)
@@ -3176,8 +3176,8 @@ cdef class IntegerMod_int64(IntegerMod_abstract):
         cdef IntegerMod_int64 lift
         cdef int_fast64_t x
 
-        import integer_mod_ring
-        lift = IntegerMod_int64(integer_mod_ring.IntegerModRing(self.__modulus.int64 * other.__modulus.int64), None, empty=True)
+        from .integer_mod_ring import IntegerModRing
+        lift = IntegerMod_int64(IntegerModRing(self.__modulus.int64 * other.__modulus.int64), None, empty=True)
 
         try:
             x = (other.ivalue - self.ivalue % other.__modulus.int64) * mod_inverse_int64(self.__modulus.int64, other.__modulus.int64)
@@ -4166,6 +4166,30 @@ cdef class IntegerMod_to_IntegerMod(IntegerMod_hom):
     def _repr_type(self):
         return "Natural"
 
+    def is_surjective(self):
+        r"""
+        Return whether this morphism is surjective.
+
+        EXAMPLES::
+
+            sage: Zmod(4).hom(Zmod(2)).is_surjective()
+            True
+
+        """
+        return True
+
+    def is_injective(self):
+        r"""
+        Return whether this morphism is injective.
+
+        EXAMPLES::
+
+            sage: Zmod(4).hom(Zmod(2)).is_injective()
+            False
+
+        """
+        return self.domain().order() == self.codomain().order()
+
 cdef class Integer_to_IntegerMod(IntegerMod_hom):
     r"""
     Fast `\ZZ \rightarrow \ZZ/n\ZZ`
@@ -4211,6 +4235,30 @@ cdef class Integer_to_IntegerMod(IntegerMod_hom):
     def section(self):
         return IntegerMod_to_Integer(self._codomain)
 
+    def is_surjective(self):
+        r"""
+        Return whether this morphism is surjective.
+
+        EXAMPLES::
+
+            sage: ZZ.hom(Zmod(2)).is_surjective()
+            True
+
+        """
+        return True
+
+    def is_injective(self):
+        r"""
+        Return whether this morphism is injective.
+
+        EXAMPLES::
+
+            sage: ZZ.hom(Zmod(2)).is_injective()
+            False
+
+        """
+        return False
+
 cdef class IntegerMod_to_Integer(Map):
     """
     Map to lift elements to :class:`~sage.rings.integer.Integer`.
diff --git a/src/sage/rings/fraction_field.py b/src/sage/rings/fraction_field.py
index 3d4aa47f98b..b8c49d230f0 100644
--- a/src/sage/rings/fraction_field.py
+++ b/src/sage/rings/fraction_field.py
@@ -1,4 +1,5 @@
-"""
+# -*- coding: utf-8 -*-
+r"""
 Fraction Field of Integral Domains
 
 AUTHORS:
@@ -8,6 +9,9 @@
 
 - Burcin Erocal
 
+- Julian Rüth (2017-06-27): embedding into the field of fractions and its
+  section
+
 EXAMPLES:
 
 Quotienting is a constructor for an element of the fraction field::
@@ -57,16 +61,12 @@
 #   Sage: System for Algebra and Geometry Experimentation
 #
 #       Copyright (C) 2005 William Stein <wstein@gmail.com>
+#                     2017 Julian Rüth <julian.rueth@fsfe.org>
 #
-#  Distributed under the terms of the GNU General Public License (GPL)
-#
-#    This code is distributed in the hope that it will be useful,
-#    but WITHOUT ANY WARRANTY; without even the implied warranty of
-#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
-#    General Public License for more details.
-#
-#  The full text of the GPL is available at:
-#
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 # ****************************************************************************
 from __future__ import absolute_import
@@ -78,9 +78,12 @@
 import sage.misc.latex as latex
 from sage.misc.cachefunc import cached_method
 
+from sage.structure.richcmp import richcmp
 from sage.structure.parent import Parent
-from sage.structure.coerce_maps import CallableConvertMap
+from sage.structure.coerce_maps import CallableConvertMap, DefaultConvertMap_unique
 from sage.categories.basic import QuotientFields
+from sage.categories.morphism import Morphism
+from sage.categories.map import Section
 
 
 def FractionField(R, names=None):
@@ -291,6 +294,10 @@ def _coerce_map_from_(self, S):
         from sage.rings.polynomial.laurent_polynomial_ring import \
             LaurentPolynomialRing_generic
 
+        if S is self._R:
+            parent = self._R.Hom(self)
+            return parent.__make_element_class__(FractionFieldEmbedding)(self._R, self, category=parent.homset_category())
+
         def wrapper(x):
             return self._element_class(self, x.numerator(), x.denominator())
 
@@ -339,7 +346,7 @@ def _number_field_to_frac_of_ring_of_integers(self, x):
 
         -  Element of ``self``
 
-        TEST:
+        TESTS:
 
         We demonstrate that :trac:`7958` is resolved in the case of
         number fields::
@@ -831,3 +838,229 @@ def class_number(self):
             1
         """
         return 1
+
+
+class FractionFieldEmbedding(DefaultConvertMap_unique):
+    r"""
+    The embedding of an integral domain into its field of fractions.
+
+    EXAMPLES::
+
+        sage: R.<x> = QQ[]
+        sage: f = R.fraction_field().coerce_map_from(R); f
+        Coercion map:
+          From: Univariate Polynomial Ring in x over Rational Field
+          To:   Fraction Field of Univariate Polynomial Ring in x over Rational Field
+ 
+    TESTS::
+
+        sage: from sage.rings.fraction_field import FractionFieldEmbedding
+        sage: isinstance(f, FractionFieldEmbedding)
+        True
+        sage: TestSuite(f).run()
+
+    Check that :trac:`23185` has been resolved::
+
+        sage: R.<x> = QQ[]
+        sage: K.<x> = FunctionField(QQ)
+        sage: R.is_subring(K)
+        True
+        sage: R.is_subring(R.fraction_field())
+        True
+
+    """
+    def is_surjective(self):
+        r"""
+        Return whether this map is surjective.
+
+        EXAMPLES::
+
+            sage: R.<x> = QQ[]
+            sage: R.fraction_field().coerce_map_from(R).is_surjective()
+            False
+
+        """
+        return self.domain().is_field()
+
+    def is_injective(self):
+        r"""
+        Return whether this map is injective.
+
+        EXAMPLES:
+
+        The map from an integral domain to its fraction field is always
+        injective:
+
+            sage: R.<x> = QQ[]
+            sage: R.fraction_field().coerce_map_from(R).is_injective()
+            True
+
+        """
+        return True
+
+    def section(self):
+        r"""
+        Return a section of this map.
+
+        EXAMPLES::
+
+            sage: R.<x> = QQ[]
+            sage: R.fraction_field().coerce_map_from(R).section()
+            Section map:
+              From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
+              To:   Univariate Polynomial Ring in x over Rational Field
+
+        """
+        from sage.categories.sets_with_partial_maps import SetsWithPartialMaps
+        from sage.all import Hom
+        parent = Hom(self.codomain(), self.domain(), SetsWithPartialMaps())
+        return parent.__make_element_class__(FractionFieldEmbeddingSection)(self)
+
+    def _richcmp_(self, other, op):
+        r"""
+        Compare this element to ``other`` with respect to ``op``.
+
+        EXAMPLES::
+
+            sage: R.<x> = QQ[]
+            sage: f = R.fraction_field().coerce_map_from(R)
+            sage: S.<y> = GF(2)[]
+            sage: g = S.fraction_field().coerce_map_from(S)
+ 
+            sage: f == g # indirect doctest
+            False
+            sage: f == f
+            True
+
+        """
+        if type(self) != type(other):
+            return NotImplemented
+        return richcmp((self.domain(), self.codomain()), (other.domain(), other.codomain()), op)
+
+    def __hash__(self):
+        r"""
+        Return a hash value for this embedding.
+
+        EXAMPLES::
+
+            sage: R.<x> = QQ[]
+            sage: hash(R.fraction_field().coerce_map_from(R)) == hash(R.fraction_field().coerce_map_from(R))
+            True
+
+        """
+        return hash((type(self), self.domain()))
+
+
+class FractionFieldEmbeddingSection(Section):
+    r"""
+    The section of the embedding of an integral domain into its field of
+    fractions.
+
+    EXAMPLES::
+
+        sage: R.<x> = QQ[]
+        sage: f = R.fraction_field().coerce_map_from(R).section(); f
+        Section map:
+          From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
+          To:   Univariate Polynomial Ring in x over Rational Field
+ 
+    TESTS::
+
+        sage: from sage.rings.fraction_field import FractionFieldEmbeddingSection
+        sage: isinstance(f, FractionFieldEmbeddingSection)
+        True
+        sage: TestSuite(f).run()
+
+    """
+    def _call_(self, x, check=True):
+        r"""
+        Evaluate this map at ``x``.
+
+        EXAMPLES::
+
+            sage: R.<x> = QQ[]
+            sage: K = R.fraction_field()
+            sage: x = K.gen()
+            sage: f = K.coerce_map_from(R).section()
+            sage: f(x)
+            x
+            sage: f(1/x)
+            Traceback (most recent call last):
+            ...
+            TypeError: fraction must have unit denominator
+
+        TESTS:
+
+        Over inexact rings, we have to take the precision of the denominators
+        into account::
+
+            sage: R=ZpCR(2)
+            sage: S.<x> = R[]
+            sage: f = x/S(R(3,absprec=2))
+            sage: S(f)
+            (1 + 2 + O(2^2))*x
+
+        """
+        if self.codomain().is_exact() and x.denominator().is_one():
+           return x.numerator()
+        if check and not x.denominator().is_unit():
+            # This should probably be a ValueError.
+            # However, too much existing code is expecting this to throw a
+            # TypeError, so we decided to keep it for the time being.
+            raise TypeError("fraction must have unit denominator")
+        return x.numerator() * x.denominator().inverse_of_unit()
+
+    def _call_with_args(self, x, args=(), kwds={}):
+        r"""
+        Evaluation this map at ``x``.
+
+        INPUT:
+
+        - ``check`` -- whether or not to check
+
+        EXAMPLES::
+
+            sage: R.<x> = QQ[]
+            sage: K = R.fraction_field()
+            sage: R(K.gen(), check=True)
+            x
+
+        """
+        check = kwds.pop('check', True)
+        if args or kwds:
+            raise NotImplementedError("__call__ can not be called with additional arguments other than check=True/False")
+        return self._call_(x, check=check)
+
+    def _richcmp_(self, other, op):
+        r"""
+        Compare this element to ``other`` with respect to ``op``.
+
+        EXAMPLES::
+
+            sage: R.<x> = QQ[]
+            sage: f = R.fraction_field().coerce_map_from(R).section()
+            sage: S.<y> = GF(2)[]
+            sage: g = S.fraction_field().coerce_map_from(S).section()
+ 
+            sage: f == g # indirect doctest
+            False
+            sage: f == f
+            True
+
+        """
+        if type(self) != type(other):
+            return NotImplemented
+        return richcmp((self.domain(), self.codomain()), (other.domain(), other.codomain()), op)
+
+    def __hash__(self):
+        r"""
+        Return a hash value for this section.
+
+        EXAMPLES::
+
+            sage: R.<x> = QQ[]
+            sage: hash(R.fraction_field().coerce_map_from(R).section()) == hash(R.fraction_field().coerce_map_from(R).section())
+            True
+
+        """
+        return hash((type(self), self.codomain()))
diff --git a/src/sage/rings/fraction_field_FpT.pxd b/src/sage/rings/fraction_field_FpT.pxd
index 01959bf54ad..b6ec99c9a62 100644
--- a/src/sage/rings/fraction_field_FpT.pxd
+++ b/src/sage/rings/fraction_field_FpT.pxd
@@ -11,6 +11,8 @@ cdef class FpTElement(RingElement):
     cdef long p
 
     cdef FpTElement _new_c(self)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cdef FpTElement _copy_c(self)
     cpdef numerator(self)
     cpdef denominator(self)
diff --git a/src/sage/rings/fraction_field_FpT.pyx b/src/sage/rings/fraction_field_FpT.pyx
index 0af4e7c8399..bb50aca733f 100644
--- a/src/sage/rings/fraction_field_FpT.pyx
+++ b/src/sage/rings/fraction_field_FpT.pyx
@@ -1105,7 +1105,7 @@ cdef class Polyring_FpT_coerce(RingHomomorphism):
             sage: f(2*t + 2, 2, reduce=False)
             (2*t + 2)/2
 
-        TEST:
+        TESTS:
 
         Check that :trac:`12217` and :trac:`16811` are fixed::
 
diff --git a/src/sage/rings/function_field/function_field.py b/src/sage/rings/function_field/function_field.py
index 72276b6481b..1f8e66e7e10 100644
--- a/src/sage/rings/function_field/function_field.py
+++ b/src/sage/rings/function_field/function_field.py
@@ -407,11 +407,11 @@ def _coerce_map_from_(self, source):
         from .function_field_order import FunctionFieldOrder
         if isinstance(source, FunctionFieldOrder):
             K = source.fraction_field()
-            source_to_K = K._generic_convert_map(source)
             if K is self:
-                return source_to_K
+                return self._generic_coerce_map(source)
+            source_to_K = K.coerce_map_from(source)
             K_to_self = self.coerce_map_from(K)
-            if K_to_self:
+            if source_to_K and K_to_self:
                 return K_to_self * source_to_K
         from sage.categories.function_fields import FunctionFields
         if source in FunctionFields():
diff --git a/src/sage/rings/homset.py b/src/sage/rings/homset.py
index 54308acec4f..36fbfe248a6 100644
--- a/src/sage/rings/homset.py
+++ b/src/sage/rings/homset.py
@@ -36,6 +36,7 @@ def is_RingHomset(H):
     """
     return isinstance(H, RingHomset_generic)
 
+
 def RingHomset(R, S, category = None):
     """
     Construct a space of homomorphisms between the rings ``R`` and ``S``.
diff --git a/src/sage/rings/integer.pxd b/src/sage/rings/integer.pxd
index c4bcfae9847..a0c7be1b0a5 100644
--- a/src/sage/rings/integer.pxd
+++ b/src/sage/rings/integer.pxd
@@ -14,6 +14,8 @@ cdef class Integer(EuclideanDomainElement):
     cpdef __pari__(self)
 
     cpdef _shift_helper(Integer self, y, int sign)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cdef _and(Integer self, Integer other)
     cdef _or(Integer self, Integer other)
     cdef _xor(Integer self, Integer other)
diff --git a/src/sage/rings/laurent_series_ring_element.pxd b/src/sage/rings/laurent_series_ring_element.pxd
index f6b2b40245d..41f2bfceb9c 100644
--- a/src/sage/rings/laurent_series_ring_element.pxd
+++ b/src/sage/rings/laurent_series_ring_element.pxd
@@ -5,4 +5,6 @@ cdef class LaurentSeries(AlgebraElement):
     cdef long __n
 
     cdef __normalize(self)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
 
diff --git a/src/sage/rings/laurent_series_ring_element.pyx b/src/sage/rings/laurent_series_ring_element.pyx
index 8ee3c4f4729..397e5bc291f 100644
--- a/src/sage/rings/laurent_series_ring_element.pyx
+++ b/src/sage/rings/laurent_series_ring_element.pyx
@@ -51,7 +51,7 @@ AUTHORS:
 
 - Robert Bradshaw: Cython version
 """
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 import operator
 
diff --git a/src/sage/rings/morphism.pyx b/src/sage/rings/morphism.pyx
index cfaa46f3484..f6418db2a48 100644
--- a/src/sage/rings/morphism.pyx
+++ b/src/sage/rings/morphism.pyx
@@ -354,10 +354,9 @@ TESTS::
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
-import ideal
-import homset
+from . import ideal
 from cpython.object cimport Py_EQ, Py_NE
 from sage.structure.richcmp cimport (richcmp, rich_to_bool,
         richcmp_not_equal)
@@ -509,7 +508,7 @@ cdef class RingMap_lift(RingMap):
         _slots['S'] = self.S
         return Morphism._extra_slots(self, _slots)
 
-    def __richcmp__(self, other, int op):
+    def _richcmp_(self, other, int op):
         """
         Compare a ring lifting maps ``self`` to ``other``.
 
@@ -615,7 +614,8 @@ cdef class RingHomomorphism(RingMap):
             True
 
         """
-        if not homset.is_RingHomset(parent):
+        from .homset import RingHomset_generic
+        if not isinstance(parent, RingHomset_generic):
             raise TypeError("parent must be a ring homset")
         RingMap.__init__(self, parent)
 
@@ -729,7 +729,7 @@ cdef class RingHomomorphism(RingMap):
                       Defn: x |--> a + b
                             y |--> a - b
                     then
-                      Conversion via FractionFieldElement map:
+                      Coercion map:
                       From: Multivariate Polynomial Ring in a, b over Rational Field
                       To:   Fraction Field of Multivariate Polynomial Ring in a, b over Rational Field
 
@@ -764,7 +764,7 @@ cdef class RingHomomorphism(RingMap):
                       From: Multivariate Polynomial Ring in x, y over Rational Field
                       To:   Multivariate Polynomial Ring in a, b over Rational Field
                     then
-                      Conversion via FractionFieldElement map:
+                      Coercion map:
                       From: Multivariate Polynomial Ring in a, b over Rational Field
                       To:   Fraction Field of Multivariate Polynomial Ring in a, b over Rational Field
 
@@ -906,7 +906,7 @@ cdef class RingHomomorphism_coercion(RingHomomorphism):
         """
         return "Ring Coercion"
 
-    def __richcmp__(self, other, int op):
+    def _richcmp_(self, other, int op):
         """
         Compare a ring coercion morphism ``self`` to ``other``.
 
@@ -1599,7 +1599,7 @@ cdef class RingHomomorphism_cover(RingHomomorphism):
         """
         return self.codomain().defining_ideal()
 
-    def __richcmp__(self, other, int op):
+    def _richcmp_(self, other, int op):
         """
         Compare ``self`` to ``other``.
 
@@ -1796,7 +1796,7 @@ cdef class RingHomomorphism_from_quotient(RingHomomorphism):
         """
         return self.phi
 
-    def __richcmp__(self, other, op):
+    def _richcmp_(self, other, op):
         """
         Compare ``self`` to ``other``.
 
diff --git a/src/sage/rings/number_field/bdd_height.py b/src/sage/rings/number_field/bdd_height.py
index 4d59cafb406..4202ee96db3 100644
--- a/src/sage/rings/number_field/bdd_height.py
+++ b/src/sage/rings/number_field/bdd_height.py
@@ -344,7 +344,7 @@ def integer_points_in_polytope(matrix, interval_radius):
 
 def bdd_height(K, height_bound, precision=53, LLL=False):
     r"""
-    Computes all elements in the number number field `K` which have relative
+    Computes all elements in the number field `K` which have relative
     multiplicative height at most ``height_bound``.
 
     The algorithm requires arithmetic with floating point numbers;
diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index f4e989efcd6..d224a344dbb 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -29,6 +29,9 @@
 
 - Peter Bruin (2016-06): make number fields fully satisfy unique representation
 
+- John Jones (2017-07): improve check for is_galois(), add is_abelian(), building on work in patch by Chris Wuthrich
+
+
 .. note::
 
    Unlike in PARI/GP, class group computations *in Sage* do *not* by default
@@ -2872,6 +2875,80 @@ def algebraic_closure(self):
         """
         return sage.rings.all.QQbar
 
+    @cached_method
+    def conductor(self, check_abelian=True):
+        r"""
+        Computes the conductor of the abelian field `K`.
+        If check_abelian is set to false and the field is not an
+        abelian extension of `\mathbb{Q}`, the output is not meaningful.
+
+        INPUT:
+
+        - ``check_abelian`` - a boolean (default: ``True``); check to see that this is an abelian extension of `\mathbb{Q}`
+
+        OUTPUT:
+
+        Integer which is the conductor of the field.
+
+        EXAMPLES::
+
+            sage: K = CyclotomicField(27)
+            sage: k = K.subfields(9)[0][0]
+            sage: k.conductor()
+            27
+            sage: K.<t> = NumberField(x^3+x^2-2*x-1)
+            sage: K.conductor()
+            7
+            sage: K.<t> = NumberField(x^3+x^2-36*x-4)
+            sage: K.conductor()
+            109
+            sage: K = CyclotomicField(48)
+            sage: k = K.subfields(16)[0][0]
+            sage: k.conductor()
+            48
+            sage: NumberField(x,'a').conductor()
+            1
+            sage: NumberField(x^8 - 8*x^6 + 19*x^4 - 12*x^2 + 1,'a').conductor()
+            40
+            sage: NumberField(x^8 + 7*x^4 + 1,'a').conductor()
+            40
+            sage: NumberField(x^8 - 40*x^6 + 500*x^4 - 2000*x^2 + 50,'a').conductor()
+            160
+
+        ALGORITHM:
+
+            For odd primes, it is easy to compute from the ramification
+            index because the p-Sylow subgroup is cyclic.  For p=2, there
+            are two choices for a given ramification index.  They can be
+            distinguished by the parity of the exponent in the discriminant
+            of a 2-adic completion.
+        """
+        m = 1
+        if check_abelian and not self.is_abelian():
+            raise ValueError("The conductor is only defined for abelian fields")
+
+        try:
+            De = self.__disc
+        except AttributeError:
+            De = self.polynomial().discriminant()
+            A = De.numerator().prime_factors()+De.denominator().prime_factors()
+        else:
+            A = De.prime_factors()
+
+        for p in A:
+            R = self.maximal_order(p)
+            e = R.fractional_ideal(p).prime_factors()[0].ramification_index()
+            if e!= 1:
+                if p==2:
+                    m *= e*2
+                    c = R.discriminant().valuation(2)
+                    c /= self.polynomial().degree()/e
+                    if is_odd(c):
+                        m *= 2
+                else:
+                    m *= p**(e.valuation(p)+1)
+        return m
+
     def latex_variable_name(self, name=None):
         """
         Return the latex representation of the variable name for this
@@ -3476,6 +3553,38 @@ def primes_of_degree_one_list(self, n, num_integer_primes=10000, max_iterations=
         it = self.primes_of_degree_one_iter()
         return [ next(it) for i in range(n) ]
 
+    def completely_split_primes(self, B = 200):
+        r"""
+        Returns a list of rational primes which split completely in the number field `K`.
+
+        INPUT:
+
+        - ``B`` -- a positive integer bound (default: 200)
+
+        OUTPUT:
+
+        A list of all primes ``p < B`` which split completely in ``K``.
+
+       EXAMPLE::
+
+            sage: K.<xi> = NumberField(x^3 - 3*x + 1)
+            sage: K.completely_split_primes(100)
+            [17, 19, 37, 53, 71, 73, 89]
+
+        """
+        from sage.rings.fast_arith import prime_range
+        from sage.rings.finite_rings.finite_field_constructor import GF
+        from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+        from sage.arith.all import factor
+        split_primes = []
+        for p in prime_range(B):
+            Fp = GF(p)
+            FpT = PolynomialRing(Fp,'T')
+            g = FpT(self.defining_polynomial())
+            if len(factor(g)) == self.degree():
+                split_primes.append(p)
+        return split_primes
+
     def _is_valid_homomorphism_(self, codomain, im_gens):
         """
         Return whether or not there is a homomorphism defined by the given
@@ -4067,7 +4176,7 @@ def S_units(self, S, proof=True):
             sage: p[0].parent()
             Number Field in alpha with defining polynomial x^3 + x + 1
 
-        TEST:
+        TESTS:
 
         This checks that the multiple entries issue at :trac:`9341` is fixed::
 
@@ -5129,8 +5238,50 @@ def is_galois(self):
             True
             sage: NumberField(x^3 + 2, 'a').is_galois()
             False
+            sage: NumberField(x^15 + x^14 - 14*x^13 - 13*x^12 + 78*x^11 + 66*x^10 - 220*x^9 - 165*x^8 + 330*x^7 + 210*x^6 - 252*x^5 - 126*x^4 + 84*x^3 + 28*x^2 - 8*x - 1, 'a').is_galois()
+            True
+            sage: NumberField(x^15 + x^14 - 14*x^13 - 13*x^12 + 78*x^11 + 66*x^10 - 220*x^9 - 165*x^8 + 330*x^7 + 210*x^6 - 252*x^5 - 126*x^4 + 84*x^3 + 28*x^2 - 8*x - 10, 'a').is_galois()
+            False
+        """
+        #return self.galois_group(type="pari").order() == self.degree()
+        if self.degree() < 12:
+            return self.galois_group(type='pari').order() == self.degree()
+        else:
+            return len(self.automorphisms()) == self.degree()
+
+    @cached_method
+    def is_abelian(self):
+        r"""
+        Return True if this number field is an abelian Galois extension of
+        `\QQ`.
+
+        EXAMPLES::
+
+            sage: NumberField(x^2 + 1, 'i').is_abelian()
+            True
+            sage: NumberField(x^3 + 2, 'a').is_abelian()
+            False
+            sage: NumberField(x^3 + x^2 - 2*x - 1, 'a').is_abelian()
+            True
+            sage: NumberField(x^6 + 40*x^3 + 1372, 'a').is_abelian()
+            False
+            sage: NumberField(x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1, 'a').is_abelian()
+            True
         """
-        return self.galois_group(type="pari").order() == self.degree()
+
+        if not self.is_galois():
+            return False
+
+        d = self.degree()
+        dsqrt = d.isqrt()
+        if d == 1 or d.is_prime() or (d == dsqrt**2 and dsqrt.is_prime()):
+            return True
+
+        if d <= 11:
+            return self.galois_group().is_abelian()
+
+        pari_pol = pari(self.polynomial())
+        return pari_pol.galoisinit().galoisisabelian(1)==1
 
     @cached_method
     def galois_group(self, type=None, algorithm='pari', names=None):
@@ -6078,7 +6229,7 @@ def units(self, proof=None):
             sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3
             sage: K = NumberField(A, 'a')
             sage: K.units()
-            (1/275*a^3 + 4/55*a^2 - 5/11*a + 3,)
+            (1/275*a^3 - 7/55*a^2 + 6/11*a - 3,)
 
         For big number fields, provably computing the unit group can
         take a very long time.  In this case, one can ask for the
@@ -10220,9 +10371,10 @@ def _multiplicative_order_table(self):
 
     def zeta(self, n=None, all=False):
         """
-        Return an element of multiplicative order `n` in this this
-        cyclotomic field. If there is no such element, raise a
-        ``ValueError``.
+        Return an element of multiplicative order `n` in this
+        cyclotomic field.
+
+        If there is no such element, raise a ``ValueError``.
 
         INPUT:
 
diff --git a/src/sage/rings/number_field/number_field_element.pxd b/src/sage/rings/number_field/number_field_element.pxd
index 0279df04d09..5f14f99559d 100644
--- a/src/sage/rings/number_field/number_field_element.pxd
+++ b/src/sage/rings/number_field/number_field_element.pxd
@@ -20,6 +20,11 @@ cdef class NumberFieldElement(FieldElement):
     cdef object __matrix
 
     cdef _new(self)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
+
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
 
     cdef number_field(self)
 
diff --git a/src/sage/rings/number_field/number_field_element.pyx b/src/sage/rings/number_field/number_field_element.pyx
index 65b4e4a472d..fdceffd9b26 100644
--- a/src/sage/rings/number_field/number_field_element.pyx
+++ b/src/sage/rings/number_field/number_field_element.pyx
@@ -176,7 +176,6 @@ def _inverse_mod_generic(elt, I):
             y += v[j] * sum([b[j,i+n] * B[i] for i in xrange(n)])
     return I.small_residue(y)
 
-__pynac_pow = False
 
 cdef class NumberFieldElement(FieldElement):
     """
diff --git a/src/sage/rings/number_field/number_field_element_quadratic.pyx b/src/sage/rings/number_field/number_field_element_quadratic.pyx
index 478cd79a09f..2a2e945d694 100644
--- a/src/sage/rings/number_field/number_field_element_quadratic.pyx
+++ b/src/sage/rings/number_field/number_field_element_quadratic.pyx
@@ -324,7 +324,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         """
         Used for pickling.
 
-        TEST:
+        TESTS:
 
             sage: K.<a> = NumberField(x^2-13)
             sage: loads(dumps(a)) == a
diff --git a/src/sage/rings/number_field/totallyreal_rel.py b/src/sage/rings/number_field/totallyreal_rel.py
index 3932d057542..e448aa17912 100644
--- a/src/sage/rings/number_field/totallyreal_rel.py
+++ b/src/sage/rings/number_field/totallyreal_rel.py
@@ -55,7 +55,7 @@
     [False, True, True, True, False, False, True, True, False, False, False, False, False, True, True, False, False, True, False, False, False]
 
 Eight out of 21 such fields are Galois (with Galois group `C_4`
-or `C_2 \times C_2`); the others have have Galois closure of degree 8
+or `C_2 \times C_2`); the others have Galois closure of degree 8
 (with Galois group `D_8`).
 
 Finally, we compute the cubic extensions of `\QQ(\zeta_7)^+` with
diff --git a/src/sage/rings/number_field/unit_group.py b/src/sage/rings/number_field/unit_group.py
index b7bcde93cc2..69b527bf80b 100644
--- a/src/sage/rings/number_field/unit_group.py
+++ b/src/sage/rings/number_field/unit_group.py
@@ -100,29 +100,29 @@
     sage: UL.zeta_order()
     24
     sage: UL.roots_of_unity()
-    [-b*a,
-     -b^2*a - b^2,
+    [b*a + b,
+     b^2*a,
      -b^3,
-     -a,
-     -b*a - b,
+     a + 1,
+     b*a,
      -b^2,
-     b^3*a,
-     -a - 1,
+     -b^3*a - b^3,
+     a,
      -b,
-     b^2*a,
-     b^3*a + b^3,
+     -b^2*a - b^2,
+     -b^3*a,
      -1,
-     b*a,
-     b^2*a + b^2,
+     -b*a - b,
+     -b^2*a,
      b^3,
-     a,
-     b*a + b,
+     -a - 1,
+     -b*a,
      b^2,
-     -b^3*a,
-     a + 1,
+     b^3*a + b^3,
+     -a,
      b,
-     -b^2*a,
-     -b^3*a - b^3,
+     b^2*a + b^2,
+     b^3*a,
      1]
 
 A relative extension example, which worked thanks to the code review by F.W.Clarke::
diff --git a/src/sage/rings/padics/CA_template.pxi b/src/sage/rings/padics/CA_template.pxi
index ab33ab87b5f..00e413e904e 100644
--- a/src/sage/rings/padics/CA_template.pxi
+++ b/src/sage/rings/padics/CA_template.pxi
@@ -495,7 +495,7 @@ cdef class CAElement(pAdicTemplateElement):
 
         INPUT:
 
-        - ``absprec`` -- an integer
+        - ``absprec`` -- an integer or infinity
 
         OUTPUT:
 
@@ -513,16 +513,34 @@ cdef class CAElement(pAdicTemplateElement):
             2 + 3 + 3^2 + 3^3 + O(3^5)
             sage: a.add_bigoh(3)
             2 + 3 + 3^2 + O(3^3)
+
+        TESTS:
+
+        Verify that :trac:`13591` has been resolved::
+
+            sage: k(3).add_bigoh(-1)
+            O(3^-1)
+
         """
         cdef long aprec, newprec
+        if absprec is infinity:
+            return self
         if isinstance(absprec, int):
             aprec = absprec
         else:
             if not isinstance(absprec, Integer):
                 absprec = Integer(absprec)
-            aprec = mpz_get_si((<Integer>absprec).value)
+            if mpz_fits_slong_p((<Integer>absprec).value) == 0:
+                if mpz_sgn((<Integer>absprec).value) == -1:
+                    raise ValueError("absprec must fit into a signed long")
+                else:
+                    aprec = self.prime_pow.ram_prec_cap
+            else:
+                aprec = mpz_get_si((<Integer>absprec).value)
         if aprec >= self.absprec:
             return self
+        if aprec < 0:
+            return self.parent().fraction_field()(self).add_bigoh(absprec)
         cdef CAElement ans = self._new_c()
         ans.absprec = aprec
         creduce(ans.value, self.value, ans.absprec, ans.prime_pow)
diff --git a/src/sage/rings/padics/CR_template.pxi b/src/sage/rings/padics/CR_template.pxi
index 14b295a5b5b..3711db79619 100644
--- a/src/sage/rings/padics/CR_template.pxi
+++ b/src/sage/rings/padics/CR_template.pxi
@@ -867,7 +867,15 @@ cdef class CRElement(pAdicTemplateElement):
         else:
             if not isinstance(absprec, Integer):
                 absprec = Integer(absprec)
-            aprec = mpz_get_si((<Integer>absprec).value)
+            if mpz_fits_slong_p((<Integer>absprec).value) == 0:
+                if mpz_sgn((<Integer>absprec).value) == -1:
+                    raise ValueError("absprec must fit into a signed long")
+                else:
+                    aprec = self.prime_pow.ram_prec_cap
+            else:
+                aprec = mpz_get_si((<Integer>absprec).value)
+        if aprec < 0 and not self.parent().is_field():
+            return self.parent().fraction_field()(self).add_bigoh(absprec)
         if aprec < self.ordp:
             ans = self._new_c()
             ans._set_inexact_zero(aprec)
@@ -2328,7 +2336,7 @@ cdef class pAdicCoercion_CR_frac_field(RingHomomorphism):
 
     def is_surjective(self):
         r"""
-        Return whether this map is injective.
+        Return whether this map is surjective.
 
         EXAMPLES::
 
diff --git a/src/sage/rings/padics/FM_template.pxi b/src/sage/rings/padics/FM_template.pxi
index 4f72ea20ee4..1ffa5e77486 100644
--- a/src/sage/rings/padics/FM_template.pxi
+++ b/src/sage/rings/padics/FM_template.pxi
@@ -402,7 +402,7 @@ cdef class FMElement(pAdicTemplateElement):
 
         INPUT:
 
-        - ``absprec`` -- an integer
+        - ``absprec`` -- an integer or infinity
 
         OUTPUT:
 
@@ -412,14 +412,36 @@ cdef class FMElement(pAdicTemplateElement):
 
             sage: R = Zp(7,4,'fixed-mod','series'); a = R(8); a.add_bigoh(1)
             1 + O(7^4)
+
+        TESTS:
+
+        We handle very large and very small values for ``absprec`` correctly::
+
+            sage: a = R(7)
+            sage: a.add_bigoh(2^1000)
+            7 + O(7^4)
+            sage: a.add_bigoh(-2^1000)
+            Traceback (most recent call last):
+            ...
+            ValueError: absprec must be at least 0
+
         """
-        cdef long aprec, newprec
+        cdef long aprec
+        if absprec is infinity:
+            return self
         if isinstance(absprec, int):
             aprec = absprec
         else:
             if not isinstance(absprec, Integer):
                 absprec = Integer(absprec)
-            aprec = mpz_get_si((<Integer>absprec).value)
+            if mpz_sgn((<Integer>absprec).value) == -1:
+                aprec = -1
+            elif mpz_fits_slong_p((<Integer>absprec).value) == 0:
+                aprec = self.prime_pow.ram_prec_cap
+            else:
+                aprec = mpz_get_si((<Integer>absprec).value)
+        if aprec < 0:
+            raise ValueError("absprec must be at least 0")
         if aprec >= self.prime_pow.prec_cap:
             return self
         cdef FMElement ans = self._new_c()
diff --git a/src/sage/rings/padics/FP_template.pxi b/src/sage/rings/padics/FP_template.pxi
index af4f0d40dbe..ceea80649e3 100644
--- a/src/sage/rings/padics/FP_template.pxi
+++ b/src/sage/rings/padics/FP_template.pxi
@@ -692,7 +692,7 @@ cdef class FPElement(pAdicTemplateElement):
 
         INPUT:
 
-        - ``absprec`` -- an integer
+        - ``absprec`` -- an integer or infinity
 
         OUTPUT:
 
@@ -717,11 +717,13 @@ cdef class FPElement(pAdicTemplateElement):
                 aprec = minusmaxordp
             else:
                 aprec = mpz_get_si((<Integer>absprec).value)
-        if aprec >= self.ordp + self.prime_pow.prec_cap:
+        if aprec >= self.ordp + self.prime_pow.ram_prec_cap:
             return self
         cdef FPElement ans = self._new_c()
         if aprec <= self.ordp:
             ans._set_exact_zero()
+            if aprec < 0:
+                return self.parent().fraction_field()(0)
         else:
             ans.ordp = self.ordp
             creduce(ans.unit, self.unit, aprec - self.ordp, ans.prime_pow)
@@ -1394,9 +1396,10 @@ cdef class pAdicCoercion_ZZ_FP(RingHomomorphism):
         """
         return self._section
 
+
 cdef class pAdicConvert_FP_ZZ(RingMap):
     """
-    The map from a floating point ring back to ZZ that returns the the smallest
+    The map from a floating point ring back to ZZ that returns the smallest
     non-negative integer approximation to its input which is accurate up to the precision.
 
     If the input is not in the closure of the image of ZZ, raises a ValueError.
diff --git a/src/sage/rings/padics/factory.py b/src/sage/rings/padics/factory.py
index d5556861e2a..6034161d066 100644
--- a/src/sage/rings/padics/factory.py
+++ b/src/sage/rings/padics/factory.py
@@ -69,8 +69,33 @@
 ext_table['u', pAdicFieldFloatingPoint] = UnramifiedExtensionFieldFloatingPoint
 #ext_table['u', pAdicRingLazy] = UnramifiedExtensionRingLazy
 
+def _default_show_prec(type, print_mode):
+    """
+    Returns the default show_prec value for a given type and printing mode.
+
+    INPUT:
 
-def get_key_base(p, prec, type, print_mode, names, ram_name, print_pos, print_sep, print_alphabet, print_max_terms, check, valid_non_lazy_types):
+    - ``type`` -- a string: ``'capped-rel'``, ``'capped-abs'``, ``'fixed-mod'`` or ``'floating-point'``
+    - ``print_mode`` -- a string: ``'series'``, ``'terse'``, ``'val-unit'``, ``'digits'``, ``'bars'``
+
+    EXAMPLES::
+
+        sage: from sage.rings.padics.factory import _default_show_prec
+        sage: _default_show_prec('floating-point', 'series')
+        False
+        sage: _default_show_prec('capped-rel', 'series')
+        True
+        sage: _default_show_prec('capped-abs', 'digits')
+        False
+    """
+    if type == 'floating-point':
+        return False
+    elif print_mode in ('series', 'terse', 'val-unit'):
+        return True
+    else:
+        return False
+
+def get_key_base(p, prec, type, print_mode, names, ram_name, print_pos, print_sep, print_alphabet, print_max_terms, show_prec, check, valid_non_lazy_types):
     """
     This implements create_key for Zp and Qp: moving it here prevents code duplication.
 
@@ -79,11 +104,13 @@ def get_key_base(p, prec, type, print_mode, names, ram_name, print_pos, print_se
     EXAMPLES::
 
         sage: from sage.rings.padics.factory import get_key_base
-        sage: get_key_base(11, 5, 'capped-rel', None, None, None, None, ':', None, None, True, ['capped-rel'])
-        (11, 5, 'capped-rel', 'series', '11', True, '|', (), -1)
-        sage: get_key_base(12, 5, 'capped-rel', 'digits', None, None, None, None, None, None, False, ['capped-rel'])
-        (12, 5, 'capped-rel', 'digits', '12', True, '|', ('0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B'), -1)
+        sage: get_key_base(11, 5, 'capped-rel', None, None, None, None, ':', None, None, False, True, ['capped-rel'])
+        (11, 5, 'capped-rel', 'series', '11', True, '|', (), -1, False)
+        sage: get_key_base(12, 5, 'capped-rel', 'digits', None, None, None, None, None, None, True, False, ['capped-rel'])
+        (12, 5, 'capped-rel', 'digits', '12', True, '|', ('0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B'), -1, True)
     """
+    if prec is None:
+        prec = DEFAULT_PREC
     if check:
         if not isinstance(p, Integer):
             p = Integer(p)
@@ -107,6 +134,8 @@ def get_key_base(p, prec, type, print_mode, names, ram_name, print_pos, print_se
             print_max_terms = print_mode['max_ram_terms']
         if 'max_terms' in print_mode:
             print_max_terms = print_mode['max_terms']
+        if 'show_prec' in print_mode:
+            show_prec = print_mode['show_prec']
         if 'mode' in print_mode:
             print_mode = print_mode['mode']
         else:
@@ -154,8 +183,10 @@ def get_key_base(p, prec, type, print_mode, names, ram_name, print_pos, print_se
             name = names
         else:
             name = str(names)
+    if show_prec is None:
+        show_prec = _default_show_prec(type, print_mode)
     if type in valid_non_lazy_types:
-        key = (p, prec, type, print_mode, name, print_pos, print_sep, tuple(print_alphabet), print_max_terms)
+        key = (p, prec, type, print_mode, name, print_pos, print_sep, tuple(print_alphabet), print_max_terms, show_prec)
     else:
         print(type)
         raise ValueError("type must be %s"%(", ".join(valid_non_lazy_types)))
@@ -210,6 +241,9 @@ class Qp_class(UniqueFactory):
     - ``print_max_terms`` -- integer (default ``None``) The maximum number of
       terms shown.  See PRINTING below.
 
+    - ``show_prec`` -- bool (default ``None``) whether to show the precision
+      for elements.  See PRINTING below.
+
     - ``check`` -- bool (default ``True``) whether to check if `p` is prime.
       Non-prime input may cause seg-faults (but can also be useful for
       base n expansions for example)
@@ -296,6 +330,11 @@ class Qp_class(UniqueFactory):
         sage: U.<p> = Qp(5); p
         p + O(p^21)
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: Qp(5, show_prec=False)(6)
+        1 + 5
+
     *print_sep* and *print_alphabet* have no effect in series mode.
 
     Note that print options affect equality::
@@ -321,6 +360,11 @@ class Qp_class(UniqueFactory):
         sage: T = Qp(5, print_mode='val-unit', names='pi'); a = T(70700); a
         pi^2 * 2828 + O(pi^22)
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: Qp(5, print_mode='val-unit', show_prec=False)(30)
+        5 * 6
+
     *print_max_terms*, *print_sep* and *print_alphabet* have no effect.
 
     Equality again depends on the printing options::
@@ -358,6 +402,11 @@ class Qp_class(UniqueFactory):
         sage: d = T(-707/5^10); d
         95367431639918/unif^10 + O(unif^10)
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: Qp(5, print_mode='terse', show_prec=False)(6)
+        6
+
     *print_max_terms*, *print_sep* and *print_alphabet* have no effect.
 
     Equality depends on printing options::
@@ -420,6 +469,11 @@ class Qp_class(UniqueFactory):
         sage: T = Qp(5, print_mode='digits', print_max_terms=4, print_alphabet=('1','2','3','4','5')); b = T(-70700); repr(b)
         '...325311'
 
+    *show_prec* determines whether the precision is printed (default ``False``)::
+
+        sage: repr(Zp(5, print_mode='digits', show_prec=True)(6))
+        '...11 + O(5^20)'
+
     *print_pos*, *name* and *print_sep* have no effect.
 
     Equality depends on printing options::
@@ -464,6 +518,11 @@ class Qp_class(UniqueFactory):
         sage: U = Qp(5, print_mode='bars', print_sep=']['); a = U(70700); repr(a)
         '...4][2][3][0][3][0][0'
 
+    *show_prec* determines whether the precision is printed (default ``False``)::
+
+        sage: repr(Zp(5, print_mode='bars', show_prec=True)(6))
+        '...1|1 + O(5^20)'
+
     *name* and *print_alphabet* have no effect.
 
     Equality depends on printing options::
@@ -476,9 +535,9 @@ class Qp_class(UniqueFactory):
         sage: K = Qp(15, check=False); a = K(999); a
         9 + 6*15 + 4*15^2 + O(15^20)
     """
-    def create_key(self, p, prec = DEFAULT_PREC, type = 'capped-rel', print_mode = None,
+    def create_key(self, p, prec = None, type = 'capped-rel', print_mode = None,
                    names = None, ram_name = None, print_pos = None,
-                   print_sep = None, print_alphabet = None, print_max_terms = None, check = True):
+                   print_sep = None, print_alphabet = None, print_max_terms = None, show_prec=None, check = True):
         """
         Creates a key from input parameters for ``Qp``.
 
@@ -487,7 +546,7 @@ def create_key(self, p, prec = DEFAULT_PREC, type = 'capped-rel', print_mode = N
         TESTS::
 
             sage: Qp.create_key(5,40)
-            (5, 40, 'capped-rel', 'series', '5', True, '|', (), -1)
+            (5, 40, 'capped-rel', 'series', '5', True, '|', (), -1, True)
         """
         if isinstance(names, (int, Integer)):
             # old pickle; names is what used to be halt.
@@ -497,7 +556,7 @@ def create_key(self, p, prec = DEFAULT_PREC, type = 'capped-rel', print_mode = N
             print_alphabet = print_max_terms
             print_max_terms = check
             check = True
-        return get_key_base(p, prec, type, print_mode, names, ram_name, print_pos, print_sep, print_alphabet, print_max_terms, check, ['capped-rel', 'floating-point'])
+        return get_key_base(p, prec, type, print_mode, names, ram_name, print_pos, print_sep, print_alphabet, print_max_terms, show_prec, check, ['capped-rel', 'floating-point'])
 
     def create_object(self, version, key):
         """
@@ -513,8 +572,11 @@ def create_object(self, version, key):
         if version[0] < 3 or (version[0] == 3 and version[1] < 2) or (version[0] == 3 and version[1] == 2 and version[2] < 3):
             p, prec, type, print_mode, name = key
             print_pos, print_sep, print_alphabet, print_max_terms = None, None, None, None
-        else:
+        elif version[0] < 8:
             p, prec, type, print_mode, name, print_pos, print_sep, print_alphabet, print_max_terms = key
+            show_prec = None
+        else:
+            p, prec, type, print_mode, name, print_pos, print_sep, print_alphabet, print_max_terms, show_prec = key
         if isinstance(type, Integer):
             # lazy
             raise NotImplementedError("lazy p-adics need more work.  Sorry.")
@@ -523,29 +585,29 @@ def create_object(self, version, key):
             # keys changed in order to reduce irrelevant duplications: e.g. two Qps with print_mode 'series'
             # that are identical except for different 'print_alphabet' now return the same object.
             key = get_key_base(p, prec, type, print_mode, name, None, print_pos, print_sep, print_alphabet,
-                               print_max_terms, False, ['capped-rel', 'fixed-mod', 'capped-abs'])
+                               print_max_terms, None, False, ['capped-rel', 'fixed-mod', 'capped-abs'])
             try:
                 obj = self._cache[version, key]()
                 if obj is not None:
                     return obj
             except KeyError:
                 pass
-            p, prec, type, print_mode, name, print_pos, print_sep, print_alphabet, print_max_terms = key
+            p, prec, type, print_mode, name, print_pos, print_sep, print_alphabet, print_max_terms, show_prec = key
 
         if type == 'capped-rel':
             if print_mode == 'terse':
                 return pAdicFieldCappedRelative(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                          'ram_name': name, 'max_terse_terms': print_max_terms}, name)
+                                                          'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name)
             else:
                 return pAdicFieldCappedRelative(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                          'ram_name': name, 'max_ram_terms': print_max_terms}, name)
+                                                          'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name)
         elif type == 'floating-point':
             if print_mode == 'terse':
                 return pAdicFieldFloatingPoint(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                          'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': False}, name)
+                                                          'ram_name': name, 'max_terse_terms': print_max_terms, 'show_prec': show_prec}, name)
             else:
                 return pAdicFieldFloatingPoint(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                         'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': False}, name)
+                                                         'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name)
         else:
             raise ValueError("unexpected type")
 
@@ -559,7 +621,7 @@ def create_object(self, version, key):
 def Qq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
           print_mode=None, ram_name = None, res_name = None, print_pos = None,
        print_sep = None, print_max_ram_terms = None,
-       print_max_unram_terms = None, print_max_terse_terms = None, check = True, implementation = 'FLINT'):
+       print_max_unram_terms = None, print_max_terse_terms = None, show_prec=None, check = True, implementation = 'FLINT'):
     """
     Given a prime power `q = p^n`, return the unique unramified
     extension of `\mathbb{Q}_p` of degree `n`.
@@ -615,6 +677,9 @@ def Qq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
       number of terms in the polynomial representation of an element
       (using ``'terse'``).  See PRINTING below.
 
+    - ``show_prec`` -- bool (default ``None``) whether to show the precision
+      for elements.  See PRINTING below.
+
     - ``check`` -- bool (default ``True``) whether to check inputs.
 
     OUTPUT:
@@ -784,6 +849,11 @@ def Qq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
         sage: V.<f> = Qq(128, prec = 8, print_mode='series', print_max_unram_terms = 0); repr((1+f)^9 - 1 - f^3)
         '(...)*2 + (...)*2^2 + (...)*2^3 + (...)*2^4 + (...)*2^5 + (...)*2^6 + (...)*2^7 + O(2^8)'
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: U.<e> = Qq(9, 2, show_prec=False); repr(-3*(1+2*e)^4)
+        '3 + e*3^2'
+
     *print_sep* and *print_max_terse_terms* have no effect.
 
     Note that print options affect equality::
@@ -822,6 +892,11 @@ def Qq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
         sage: b*17*(a^3-a+14)
         1 + O(17^6)
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: U.<e> = Qq(9, 2, print_mode='val-unit', show_prec=False); repr(-3*(1+2*e)^4)
+        '3 * (1 + 3*e)'
+
     *print_sep*, *print_max_ram_terms* and *print_max_unram_terms* have no
     effect.
 
@@ -867,6 +942,11 @@ def Qq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
         sage: U.<a> = Qq(625, print_mode='terse', print_max_terse_terms=2); (a-1/5)^6
         106251/5^6 + 49994/5^5*a + ... + O(5^14)
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: U.<e> = Qq(9, 2, print_mode='terse', show_prec=False); repr(-3*(1+2*e)^4)
+        '3 + 9*e'
+
     *print_sep*, *print_max_ram_terms* and *print_max_unram_terms* have no
     effect.
 
@@ -955,6 +1035,11 @@ def Qq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
         sage: with local_print_mode(U, {'max_unram_terms':0}): repr(b-75*a)
         '...[...][...][...][...][][...][...]'
 
+    *show_prec* determines whether the precision is printed (default ``False``)::
+
+        sage: U.<e> = Qq(9, 2, print_mode='bars', show_prec=True); repr(-3*(1+2*e)^4)
+        '...[0, 1]|[1]|[] + O(3^3)'
+
     *ram_name* and *print_max_terse_terms* have no effect.
 
     Equality depends on printing options::
@@ -1028,8 +1113,8 @@ def Qq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
     if prec is not None and not isinstance(prec, Integer):
         prec = Integer(prec)
 
-    base = Qp(p=p, prec=DEFAULT_PREC if prec is None else prec, type=type, print_mode=print_mode, names=ram_name, print_pos=print_pos,
-              print_sep=print_sep, print_max_terms=print_max_ram_terms, check=check)
+    base = Qp(p=p, prec=prec, type=type, print_mode=print_mode, names=ram_name, print_pos=print_pos,
+              print_sep=print_sep, print_max_terms=print_max_ram_terms, show_prec=show_prec, check=check)
 
     if k == 1:
         return base
@@ -1053,15 +1138,14 @@ def Qq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
                             names=names, res_name=res_name, ram_name=ram_name, print_pos=print_pos,
                             print_sep=print_sep, print_max_ram_terms=print_max_ram_terms,
                             print_max_unram_terms=print_max_unram_terms,
-                            print_max_terse_terms=print_max_terse_terms, check=check,
+                            print_max_terse_terms=print_max_terse_terms, show_prec=show_prec, check=check,
                             unram=True, implementation=implementation)
 
 ######################################################
 # Short constructor names for different types
 ######################################################
 
-def QpCR(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = None,
-         print_sep = None, print_alphabet = None, print_max_terms = None, check=True):
+def QpCR(p, prec = None, *args, **kwds):
     """
     A shortcut function to create capped relative `p`-adic fields.
 
@@ -1073,12 +1157,9 @@ def QpCR(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = No
         sage: QpCR(5, 40)
         5-adic Field with capped relative precision 40
     """
-    return Qp(p=p, prec=prec, print_mode=print_mode, check=check, names=names,
-              print_pos=print_pos, print_sep=print_sep, print_alphabet=print_alphabet, print_max_terms=print_max_terms,
-              type = 'capped-rel')
+    return Qp(p, prec, 'capped-rel', *args, **kwds)
 
-def QpFP(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = None,
-         print_sep = None, print_alphabet = None, print_max_terms = None, check=True):
+def QpFP(p, prec = None, *args, **kwds):
     """
     A shortcut function to create floating point `p`-adic fields.
 
@@ -1090,14 +1171,9 @@ def QpFP(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = No
         sage: QpFP(5, 40)
         5-adic Field with floating precision 40
     """
-    return Qp(p=p, prec=prec, print_mode=print_mode, check=check, names=names,
-              print_pos=print_pos, print_sep=print_sep, print_alphabet=print_alphabet, print_max_terms=print_max_terms,
-              type = 'floating-point')
-
-def QqCR(q, prec = None, modulus = None, names=None,
-          print_mode=None, ram_name = None, print_pos = None,
-       print_sep = None, print_alphabet = None, print_max_ram_terms = None,
-       print_max_unram_terms = None, print_max_terse_terms = None, check = True, implementation = 'FLINT'):
+    return Qp(p, prec, 'floating-point', *args, **kwds)
+
+def QqCR(q, prec = None, *args, **kwds):
     """
     A shortcut function to create capped relative unramified `p`-adic
     fields.
@@ -1110,15 +1186,9 @@ def QqCR(q, prec = None, modulus = None, names=None,
         sage: R.<a> = QqCR(25, 40); R
         Unramified Extension in a defined by x^2 + 4*x + 2 with capped relative precision 40 over 5-adic Field
     """
-    return Qq(q, prec=prec, modulus=modulus, names=names, print_mode=print_mode,
-              ram_name=ram_name, print_pos=print_pos, print_max_ram_terms=print_max_ram_terms,
-              print_max_unram_terms=print_max_unram_terms, print_max_terse_terms=print_max_terse_terms,
-              check=check, implementation=implementation, type = 'capped-rel')
-
-def QqFP(q, prec = None, modulus = None, names=None,
-          print_mode=None, ram_name = None, print_pos = None,
-       print_sep = None, print_alphabet = None, print_max_ram_terms = None,
-       print_max_unram_terms = None, print_max_terse_terms = None, check = True, implementation = 'FLINT'):
+    return Qq(q, prec, 'capped-rel', *args, **kwds)
+
+def QqFP(q, prec = None, *args, **kwds):
     """
     A shortcut function to create floating point unramified `p`-adic
     fields.
@@ -1131,10 +1201,7 @@ def QqFP(q, prec = None, modulus = None, names=None,
         sage: R.<a> = QqFP(25, 40); R
         Unramified Extension in a defined by x^2 + 4*x + 2 with floating precision 40 over 5-adic Field
     """
-    return Qq(q, prec=prec, modulus=modulus, names=names, print_mode=print_mode,
-              ram_name=ram_name, print_pos=print_pos, print_max_ram_terms=print_max_ram_terms,
-              print_max_unram_terms=print_max_unram_terms, print_max_terse_terms=print_max_terse_terms,
-              check=check, implementation=implementation, type = 'floating-point')
+    return Qq(q, prec, 'floating-point', *args, **kwds)
 
 #######################################################################################################
 #
@@ -1183,6 +1250,9 @@ class Zp_class(UniqueFactory):
     - ``print_max_terms`` -- integer (default ``None``) The maximum
       number of terms shown.  See PRINTING below.
 
+    - ``show_prec`` -- bool (default ``None``) whether to show the precision
+      for elements.  See PRINTING below.
+
     - ``check`` -- bool (default ``True``) whether to check if `p` is
       prime.  Non-prime input may cause seg-faults (but can also be
       useful for base `n` expansions for example)
@@ -1298,6 +1368,11 @@ class Zp_class(UniqueFactory):
         sage: U.<p> = Zp(5); p
         p + O(p^21)
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: Zp(5, show_prec=False)(6)
+        1 + 5
+
     *print_sep* and *print_alphabet* have no effect.
 
     Note that print options affect equality::
@@ -1323,6 +1398,11 @@ class Zp_class(UniqueFactory):
         sage: T = Zp(5, print_mode='val-unit', names='pi'); a = T(70700); a
         pi^2 * 2828 + O(pi^22)
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: Zp(5, print_mode='val-unit', show_prec=False)(30)
+        5 * 6
+
     *print_max_terms*, *print_sep* and *print_alphabet* have no effect.
 
     Equality again depends on the printing options::
@@ -1348,6 +1428,11 @@ class Zp_class(UniqueFactory):
         sage: T.<unif> = Zp(5, print_mode='terse'); c = T(-707); c
         95367431639918 + O(unif^20)
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: Zp(5, print_mode='terse', show_prec=False)(30)
+        30
+
     *print_max_terms*, *print_sep* and *print_alphabet* have no effect.
 
     Equality depends on printing options::
@@ -1381,6 +1466,11 @@ class Zp_class(UniqueFactory):
         sage: T = Zp(5, print_mode='digits', print_max_terms=4, print_alphabet=('1','2','3','4','5')); b = T(-70700); repr(b)
         '...325311'
 
+    *show_prec* determines whether the precision is printed (default ``False``)::
+
+        sage: repr(Zp(5, 2, print_mode='digits', show_prec=True)(6))
+        '...11 + O(5^2)'
+
     *print_pos*, *name* and *print_sep* have no effect.
 
     Equality depends on printing options::
@@ -1414,6 +1504,11 @@ class Zp_class(UniqueFactory):
         sage: U = Zp(5, print_mode='bars', print_sep=']['); a = U(70700); repr(a)
         '...4][2][3][0][3][0][0'
 
+    *show_prec* determines whether the precision is printed (default ``False``)::
+
+        sage: repr(Zp(5, 2, print_mode='bars', show_prec=True)(6))
+        '1|1 + O(5^2)'
+
     *name* and *print_alphabet* have no effect.
 
     Equality depends on printing options::
@@ -1498,9 +1593,9 @@ class Zp_class(UniqueFactory):
         sage: a + b
         1 + 5 + O(5^10)
     """
-    def create_key(self, p, prec = DEFAULT_PREC, type = 'capped-rel', print_mode = None,
+    def create_key(self, p, prec = None, type = 'capped-rel', print_mode = None,
                    names = None, ram_name = None, print_pos = None, print_sep = None, print_alphabet = None,
-                   print_max_terms = None, check = True):
+                   print_max_terms = None, show_prec = None, check = True):
         """
         Creates a key from input parameters for ``Zp``.
 
@@ -1509,9 +1604,9 @@ def create_key(self, p, prec = DEFAULT_PREC, type = 'capped-rel', print_mode = N
         TESTS::
 
             sage: Zp.create_key(5,40)
-            (5, 40, 'capped-rel', 'series', '5', True, '|', (), -1)
+            (5, 40, 'capped-rel', 'series', '5', True, '|', (), -1, True)
             sage: Zp.create_key(5,40,print_mode='digits')
-            (5, 40, 'capped-rel', 'digits', '5', True, '|', ('0', '1', '2', '3', '4'), -1)
+            (5, 40, 'capped-rel', 'digits', '5', True, '|', ('0', '1', '2', '3', '4'), -1, False)
         """
         if isinstance(names, (int, Integer)):
             # old pickle; names is what used to be halt.
@@ -1522,7 +1617,7 @@ def create_key(self, p, prec = DEFAULT_PREC, type = 'capped-rel', print_mode = N
             print_max_terms = check
             check = True
         return get_key_base(p, prec, type, print_mode, names, ram_name, print_pos, print_sep, print_alphabet,
-                            print_max_terms, check, ['capped-rel', 'fixed-mod', 'capped-abs', 'floating-point'])
+                            print_max_terms, show_prec, check, ['capped-rel', 'fixed-mod', 'capped-abs', 'floating-point'])
 
     def create_object(self, version, key):
         """
@@ -1539,8 +1634,11 @@ def create_object(self, version, key):
             (len(version) > 2 and version[0] == 3 and version[1] == 2 and version[2] < 3)):
             p, prec, type, print_mode, name = key
             print_pos, print_sep, print_alphabet, print_max_terms = None, None, None, None
-        else:
+        elif version[0] < 8:
             p, prec, type, print_mode, name, print_pos, print_sep, print_alphabet, print_max_terms = key
+            show_prec = None
+        else:
+            p, prec, type, print_mode, name, print_pos, print_sep, print_alphabet, print_max_terms, show_prec = key
         if isinstance(type, Integer):
             # lazy
             raise NotImplementedError("lazy p-adics need more work.  Sorry.")
@@ -1549,26 +1647,26 @@ def create_object(self, version, key):
             # keys changed in order to reduce irrelevant duplications: e.g. two Zps with print_mode 'series'
             # that are identical except for different 'print_alphabet' now return the same object.
             key = get_key_base(p, prec, type, print_mode, name, None, print_pos, print_sep, print_alphabet,
-                               print_max_terms, False, ['capped-rel', 'fixed-mod', 'capped-abs'])
+                               print_max_terms, None, False, ['capped-rel', 'fixed-mod', 'capped-abs'])
             try:
                 obj = self._cache[version, key]()
                 if obj is not None:
                     return obj
             except KeyError:
                 pass
-            p, prec, type, print_mode, name, print_pos, print_sep, print_alphabet, print_max_terms = key
+            p, prec, type, print_mode, name, print_pos, print_sep, print_alphabet, print_max_terms, show_prec = key
         if type == 'capped-rel':
             return pAdicRingCappedRelative(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                     'ram_name': name, 'max_ram_terms': print_max_terms}, name)
+                                                     'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name)
         elif type == 'fixed-mod':
             return pAdicRingFixedMod(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                               'ram_name': name, 'max_ram_terms': print_max_terms}, name)
+                                               'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name)
         elif type == 'capped-abs':
             return pAdicRingCappedAbsolute(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                     'ram_name': name, 'max_ram_terms': print_max_terms}, name)
+                                                     'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name)
         elif type == 'floating-point':
             return pAdicRingFloatingPoint(p, prec, {'mode': print_mode, 'pos': print_pos, 'sep': print_sep, 'alphabet': print_alphabet,
-                                                     'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': False}, name)
+                                                     'ram_name': name, 'max_ram_terms': print_max_terms, 'show_prec': show_prec}, name)
         else:
             raise ValueError("unexpected type")
 
@@ -1582,7 +1680,7 @@ def create_object(self, version, key):
 def Zq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
           print_mode=None, ram_name = None, res_name = None, print_pos = None,
        print_sep = None, print_max_ram_terms = None,
-       print_max_unram_terms = None, print_max_terse_terms = None, check = True, implementation = 'FLINT'):
+       print_max_unram_terms = None, print_max_terse_terms = None, show_prec = None, check = True, implementation = 'FLINT'):
     """
     Given a prime power `q = p^n`, return the unique unramified
     extension of `\mathbb{Z}_p` of degree `n`.
@@ -1638,6 +1736,9 @@ def Zq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
       number of terms in the polynomial representation of an element
       (using ``'terse'``).  See PRINTING below.
 
+    - ``show_prec`` -- bool (default ``None``) Whether to show the precision
+      for elements.  See PRINTING below.
+
     - ``check`` -- bool (default ``True``) whether to check inputs.
 
     - ``implementation`` -- string (default ``FLINT``) which
@@ -1849,6 +1950,11 @@ def Zq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
         sage: V.<f> = Zq(128, prec = 8, print_mode='series', print_max_unram_terms = 0); repr((1+f)^9 - 1 - f^3)
         '(...)*2 + (...)*2^2 + (...)*2^3 + (...)*2^4 + (...)*2^5 + (...)*2^6 + (...)*2^7 + O(2^8)'
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: U.<e> = Zq(9, 2, show_prec=False); repr(-3*(1+2*e)^4)
+        '3 + e*3^2'
+
     *print_sep* and *print_max_terse_terms* have no effect.
 
     Note that print options affect equality::
@@ -1885,6 +1991,11 @@ def Zq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
         sage: U.<a> = Zq(17^4, 6, print_mode='val-unit', print_max_terse_terms=3); b = (17*(a^3-a+14)^6); b
         17 * (12131797 + 12076378*a + 10809706*a^2 + ...) + O(17^7)
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: U.<e> = Zq(9, 2, print_mode='val-unit', show_prec=False); repr(-3*(1+2*e)^4)
+        '3 * (1 + 3*e)'
+
     *print_sep*, *print_max_ram_terms* and *print_max_unram_terms* have no effect.
 
     Equality again depends on the printing options::
@@ -1933,6 +2044,11 @@ def Zq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
         sage: U.<a> = Zq(625, print_mode='terse', print_max_terse_terms=2); (a-1/5)^6
         106251/5^6 + 49994/5^5*a + ... + O(5^14)
 
+    *show_prec* determines whether the precision is printed (default ``True``)::
+
+        sage: U.<e> = Zq(9, 2, print_mode='terse', show_prec=False); repr(-3*(1+2*e)^4)
+        '3 + 9*e'
+
     *print_sep*, *print_max_ram_terms* and *print_max_unram_terms* have no
     effect.
 
@@ -2006,6 +2122,11 @@ def Zq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
         sage: with local_print_mode(U, {'max_unram_terms':0}): repr(b-75*a)
         '...[...][...][...][...][][...][...]'
 
+    *show_prec* determines whether the precision is printed (default ``False``)::
+
+        sage: U.<e> = Zq(9, 2, print_mode='bars', show_prec=True); repr(-3*(1+2*e)^4)
+        '[0, 1]|[1]|[] + O(3^3)'
+
     *ram_name* and *print_max_terse_terms* have no effect.
 
     Equality depends on printing options::
@@ -2081,8 +2202,9 @@ def Zq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
     else:
         F = q
         q = F[0][0]**F[0][1]
-    base = Zp(p=F[0][0], prec=DEFAULT_PREC if prec is None else prec, type=type, print_mode=print_mode, names=ram_name,
-              print_pos=print_pos, print_sep=print_sep, print_max_terms=print_max_ram_terms, check=False)
+    base = Zp(p=F[0][0], prec=prec, type=type, print_mode=print_mode, names=ram_name,
+              print_pos=print_pos, print_sep=print_sep, print_max_terms=print_max_ram_terms,
+              show_prec=show_prec, check=False)
     if F[0][1] == 1:
         return base
     elif names is None:
@@ -2098,15 +2220,14 @@ def Zq(q, prec = None, type = 'capped-rel', modulus = None, names=None,
                             names=names, res_name=res_name, ram_name=ram_name, print_pos=print_pos,
                             print_sep=print_sep, print_max_ram_terms=print_max_ram_terms,
                             print_max_unram_terms=print_max_unram_terms,
-                            print_max_terse_terms=print_max_terse_terms, check=check,
+                            print_max_terse_terms=print_max_terse_terms, show_prec=show_prec, check=check,
                             unram=True, implementation=implementation)
 
 ######################################################
 # Short constructor names for different types
 ######################################################
 
-def ZpCR(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = None,
-         print_sep = None, print_alphabet = None, print_max_terms = None, check=True):
+def ZpCR(p, prec = None, *args, **kwds):
     """
     A shortcut function to create capped relative `p`-adic rings.
 
@@ -2118,12 +2239,9 @@ def ZpCR(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = No
         sage: ZpCR(5, 40)
         5-adic Ring with capped relative precision 40
     """
-    return Zp(p=p, prec=prec, print_mode=print_mode, check=check, names=names,
-              print_pos=print_pos, print_sep=print_sep, print_alphabet=print_alphabet, print_max_terms=print_max_terms,
-              type = 'capped-rel')
+    return Zp(p, prec, 'capped-rel', *args, **kwds)
 
-def ZpCA(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = None,
-         print_sep = None, print_alphabet = None, print_max_terms = None, check=True):
+def ZpCA(p, prec = None, *args, **kwds):
     """
     A shortcut function to create capped absolute `p`-adic rings.
 
@@ -2134,12 +2252,9 @@ def ZpCA(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = No
         sage: ZpCA(5, 40)
         5-adic Ring with capped absolute precision 40
     """
-    return Zp(p=p, prec=prec, print_mode=print_mode, check=check, names=names,
-              print_pos=print_pos, print_sep=print_sep, print_alphabet=print_alphabet, print_max_terms=print_max_terms,
-              type = 'capped-abs')
+    return Zp(p, prec, 'capped-abs', *args, **kwds)
 
-def ZpFM(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = None,
-         print_sep = None, print_alphabet = None, print_max_terms = None, check=True):
+def ZpFM(p, prec = None, *args, **kwds):
     """
     A shortcut function to create fixed modulus `p`-adic rings.
 
@@ -2150,12 +2265,9 @@ def ZpFM(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = No
         sage: ZpFM(5, 40)
         5-adic Ring of fixed modulus 5^40
     """
-    return Zp(p=p, prec=prec, print_mode=print_mode, check=check, names=names,
-              print_pos=print_pos, print_sep=print_sep, print_alphabet=print_alphabet, print_max_terms=print_max_terms,
-              type = 'fixed-mod')
+    return Zp(p, prec, 'fixed-mod', *args, **kwds)
 
-def ZpFP(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = None,
-         print_sep = None, print_alphabet = None, print_max_terms = None, check=True):
+def ZpFP(p, prec = None, *args, **kwds):
     """
     A shortcut function to create floating point `p`-adic rings.
 
@@ -2167,14 +2279,9 @@ def ZpFP(p, prec = DEFAULT_PREC, print_mode = None, names = None, print_pos = No
         sage: ZpFP(5, 40)
         5-adic Ring with floating precision 40
     """
-    return Zp(p=p, prec=prec, print_mode=print_mode, check=check, names=names,
-              print_pos=print_pos, print_sep=print_sep, print_alphabet=print_alphabet, print_max_terms=print_max_terms,
-              type = 'floating-point')
-
-def ZqCR(q, prec = None, modulus = None, names=None,
-          print_mode=None, ram_name = None, print_pos = None,
-       print_sep = None, print_alphabet = None, print_max_ram_terms = None,
-       print_max_unram_terms = None, print_max_terse_terms = None, check = True, implementation = 'FLINT'):
+    return Zp(p, prec, 'floating-point', *args, **kwds)
+
+def ZqCR(q, prec = None, *args, **kwds):
     """
     A shortcut function to create capped relative unramified `p`-adic rings.
 
@@ -2186,15 +2293,9 @@ def ZqCR(q, prec = None, modulus = None, names=None,
         sage: R.<a> = ZqCR(25, 40); R
         Unramified Extension in a defined by x^2 + 4*x + 2 with capped relative precision 40 over 5-adic Ring
     """
-    return Zq(q, prec=prec, modulus=modulus, names=names, print_mode=print_mode,
-              ram_name=ram_name, print_pos=print_pos, print_max_ram_terms=print_max_ram_terms,
-              print_max_unram_terms=print_max_unram_terms, print_max_terse_terms=print_max_terse_terms,
-              check=check, implementation=implementation, type = 'capped-rel')
-
-def ZqCA(q, prec = None, modulus = None, names=None,
-          print_mode=None, ram_name = None, print_pos = None,
-       print_sep = None, print_alphabet = None, print_max_ram_terms = None,
-       print_max_unram_terms = None, print_max_terse_terms = None, check = True, implementation='FLINT'):
+    return Zq(q, prec, 'capped-rel', *args, **kwds)
+
+def ZqCA(q, prec = None, *args, **kwds):
     """
     A shortcut function to create capped absolute unramified `p`-adic rings.
 
@@ -2205,15 +2306,9 @@ def ZqCA(q, prec = None, modulus = None, names=None,
         sage: R.<a> = ZqCA(25, 40); R
         Unramified Extension in a defined by x^2 + 4*x + 2 with capped absolute precision 40 over 5-adic Ring
     """
-    return Zq(q, prec=prec, modulus=modulus, names=names, print_mode=print_mode,
-              ram_name=ram_name, print_pos=print_pos, print_max_ram_terms=print_max_ram_terms,
-              print_max_unram_terms=print_max_unram_terms, print_max_terse_terms=print_max_terse_terms,
-              check=check, implementation=implementation, type = 'capped-abs')
-
-def ZqFM(q, prec = None, modulus = None, names=None,
-          print_mode=None, ram_name = None, print_pos = None,
-       print_sep = None, print_alphabet = None, print_max_ram_terms = None,
-       print_max_unram_terms = None, print_max_terse_terms = None, check = True, implementation='FLINT'):
+    return Zq(q, prec, 'capped-abs', *args, **kwds)
+
+def ZqFM(q, prec = None, *args, **kwds):
     """
     A shortcut function to create fixed modulus unramified `p`-adic rings.
 
@@ -2224,15 +2319,9 @@ def ZqFM(q, prec = None, modulus = None, names=None,
         sage: R.<a> = ZqFM(25, 40); R
         Unramified Extension in a defined by x^2 + 4*x + 2 of fixed modulus 5^40 over 5-adic Ring
     """
-    return Zq(q, prec=prec, modulus=modulus, names=names, print_mode=print_mode,
-              ram_name=ram_name, print_pos=print_pos, print_max_ram_terms=print_max_ram_terms,
-              print_max_unram_terms=print_max_unram_terms, print_max_terse_terms=print_max_terse_terms,
-              check=check, implementation=implementation, type = 'fixed-mod')
-
-def ZqFP(q, prec = None, modulus = None, names=None,
-          print_mode=None, ram_name = None, print_pos = None,
-       print_sep = None, print_alphabet = None, print_max_ram_terms = None,
-       print_max_unram_terms = None, print_max_terse_terms = None, check = True, implementation = 'FLINT'):
+    return Zq(q, prec, 'fixed-mod', *args, **kwds)
+
+def ZqFP(q, prec = None, *args, **kwds):
     """
     A shortcut function to create floating point unramified `p`-adic rings.
 
@@ -2244,10 +2333,7 @@ def ZqFP(q, prec = None, modulus = None, names=None,
         sage: R.<a> = ZqFP(25, 40); R
         Unramified Extension in a defined by x^2 + 4*x + 2 with floating precision 40 over 5-adic Ring
     """
-    return Zq(q, prec=prec, modulus=modulus, names=names, print_mode=print_mode,
-              ram_name=ram_name, print_pos=print_pos, print_max_ram_terms=print_max_ram_terms,
-              print_max_unram_terms=print_max_unram_terms, print_max_terse_terms=print_max_terse_terms,
-              check=check, implementation=implementation, type = 'floating-point')
+    return Zq(q, prec, 'floating-point', *args, **kwds)
 
 #######################################################################################################
 #
@@ -2274,7 +2360,7 @@ def create_key_and_extra_args(self, base, modulus, prec = None, print_mode = Non
                                   unram_name = None, ram_name = None, print_pos = None,
                                   print_sep = None, print_alphabet = None, print_max_ram_terms = None,
                                   print_max_unram_terms = None, print_max_terse_terms = None,
-                                  check = True, unram = False, implementation='FLINT'):
+                                  show_prec = None, check = True, unram = False, implementation='FLINT'):
         """
         Creates a key from input parameters for pAdicExtension.
 
@@ -2287,7 +2373,7 @@ def create_key_and_extra_args(self, base, modulus, prec = None, print_mode = Non
             sage: pAdicExtension.create_key_and_extra_args(R, x^4-15,names='w')
             (('e', 5-adic Ring with capped relative precision 3,
               x^4 - 15, ('w', None, None, 'w'),
-              12, 'series', True, '|', (), -1, -1, -1, 'NTL'),
+              12, 'series', True, '|', (), -1, -1, -1, True, 'NTL'),
              {'approx_modulus': (1 + O(5^3))*x^4 + (O(5^4))*x^3 + (O(5^4))*x^2 + (O(5^4))*x + (2*5 + 4*5^2 + 4*5^3 + O(5^4))})
 
             sage: A = Qp(3,5)
@@ -2320,6 +2406,8 @@ def create_key_and_extra_args(self, base, modulus, prec = None, print_mode = Non
             print_max_unram_terms = base._printer._max_unram_terms()
         if print_max_terse_terms is None:
             print_max_terse_terms = base._printer._max_terse_terms()
+        if show_prec is None:
+            show_prec = _default_show_prec(base._prec_type(), print_mode)
         from sage.symbolic.expression import is_Expression
         if check:
             if is_Expression(modulus):
@@ -2395,7 +2483,7 @@ def create_key_and_extra_args(self, base, modulus, prec = None, print_mode = Non
             implementation = "NTL" # for testing - FLINT ramified extensions not implemented yet
         key = (polytype, base, exact_modulus, names, prec, print_mode, print_pos,
                print_sep, tuple(print_alphabet), print_max_ram_terms, print_max_unram_terms,
-               print_max_terse_terms, implementation)
+               print_max_terse_terms, show_prec, implementation)
         return key, {'approx_modulus': approx_modulus}
 
     def create_object(self, version, key, approx_modulus=None, shift_seed=None):
@@ -2423,17 +2511,19 @@ def create_object(self, version, key, approx_modulus=None, shift_seed=None):
                 exact_modulus = premodulus.polynomial(base.exact_field())
             elif is_Polynomial(premodulus):
                 exact_modulus = premodulus.change_ring(base.exact_field())
+            show_prec = None
         else:
             (polytype, base, exact_modulus, names, prec, print_mode, print_pos,
              print_sep, print_alphabet, print_max_ram_terms, print_max_unram_terms,
-             print_max_terse_terms, implementation) = key
+             print_max_terse_terms, show_prec, implementation) = key
             if polytype == 'e':
                 unif = exact_modulus.base_ring()(base.uniformizer())
                 shift_seed = (-exact_modulus[:exact_modulus.degree()] / unif).change_ring(base)
             if not krasner_check(exact_modulus, prec):
                 raise ValueError("polynomial does not determine a unique extension.  Please specify more precision or use parameter check=False.")
 
-        show_prec = base._printer._show_prec()
+        if show_prec is None:
+            show_prec = base._printer._show_prec()
         if polytype == 'p':
             raise NotImplementedError("Extensions by general polynomials not yet supported.  Please use an unramified or Eisenstein polynomial.")
         T = ext_table[polytype, type(base.ground_ring_of_tower()).__base__]
diff --git a/src/sage/rings/padics/local_generic.py b/src/sage/rings/padics/local_generic.py
index 4eb4d62d332..dd4fa3746d8 100644
--- a/src/sage/rings/padics/local_generic.py
+++ b/src/sage/rings/padics/local_generic.py
@@ -20,11 +20,13 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
+from copy import copy
 from sage.rings.ring import CommutativeRing
 from sage.categories.complete_discrete_valuation import CompleteDiscreteValuationRings, CompleteDiscreteValuationFields
 from sage.structure.category_object import check_default_category
 from sage.structure.parent import Parent
 from sage.rings.integer import Integer
+from sage.rings.integer_ring import ZZ
 
 class LocalGeneric(CommutativeRing):
     def __init__(self, base, prec, names, element_class, category=None):
@@ -190,6 +192,278 @@ def _latex_(self):
         """
         return self._repr_(do_latex = True)
 
+    def change(self, **kwds):
+        r"""
+        Return a new ring with changed attributes.
+
+        INPUT:
+
+        The following arguments are applied to every ring in the tower:
+
+        - ``type`` -- string, the precision type
+        - ``p`` -- the prime of the ground ring.  Defining polynomials
+                   will be converted to the new base rings.
+        - ``print_mode`` -- string
+        - ``print_pos`` -- bool
+        - ``print_sep`` -- string
+        - ``print_alphabet`` -- dict
+        - ``show_prec`` -- bool
+        - ``check`` -- bool
+
+        The following arguments are only applied to the top ring in the tower:
+
+        - ``var_name`` -- string
+        - ``res_name`` -- string
+        - ``unram_name`` -- string
+        - ``ram_name`` -- string
+        - ``names`` -- string
+        - ``modulus`` -- polynomial
+
+        The following arguments have special behavior:
+
+        - ``prec`` -- integer.  If the precision is increased on an extension ring,
+                       the precision on the base is increased as necessary (respecting ramification).
+                       If the precision is decreased, the precision of the base is unchanged.
+
+        - ``field`` -- bool.  If ``True``, switch to a tower of fields via the fraction field.
+                        If False, switch to a tower of rings of integers.
+
+        - ``q`` -- prime power.  Replace the initial unramified extension of `\QQ_p` or `\ZZ_p`
+                    with an unramified extension of residue cardinality `q`.
+                    If the initial extension is ramified, add in an unramified extension.
+
+        - ``base`` -- ring or field. Use a specific base ring instead of recursively
+                       calling :meth:`change` down the tower.
+
+        See the :mod:`constructors <sage.rings.padics.factory>` for more details on the
+        meaning of these arguments.
+
+        EXAMPLES:
+
+        We can use this method to change the precision::
+
+            sage: Zp(5).change(prec=40)
+            5-adic Ring with capped relative precision 40
+
+        or the precision type::
+
+            sage: Zp(5).change(type="capped-abs")
+            5-adic Ring with capped absolute precision 20
+
+        or even the prime::
+
+            sage: ZpCA(3).change(p=17)
+            17-adic Ring with capped absolute precision 20
+
+        You can switch between the ring of integers and its fraction field::
+
+            sage: ZpCA(3).change(field=True)
+            3-adic Field with capped relative precision 20
+
+        You can also change print modes::
+
+            sage: R = Zp(5).change(prec=5, print_mode='digits')
+            sage: repr(~R(17))
+            '...13403'
+
+        Changing print mode to 'digits' works for Eisenstein extensions::
+
+            sage: S.<x> = ZZ[]
+            sage: W.<w> = Zp(3).extension(x^4 + 9*x^2 + 3*x - 3)
+            sage: W.print_mode()
+            'series'
+            sage: W.change(print_mode='digits').print_mode()
+            'digits'
+
+        You can change extensions::
+
+            sage: K.<a> = QqFP(125, prec=4)
+            sage: K.change(q=64)
+            Unramified Extension in a defined by x^6 + x^4 + x^3 + x + 1 with floating precision 4 over 2-adic Field
+            sage: R.<x> = QQ[]
+            sage: K.change(modulus = x^2 - x + 2, print_pos=False)
+            Unramified Extension in a defined by x^2 - x + 2 with floating precision 4 over 5-adic Field
+
+        and variable names::
+
+            sage: K.change(names='b')
+            Unramified Extension in b defined by x^3 + 3*x + 3 with floating precision 4 over 5-adic Field
+
+        and precision::
+
+            sage: Kup = K.change(prec=8); Kup
+            Unramified Extension in a defined by x^3 + 3*x + 3 with floating precision 8 over 5-adic Field
+            sage: Kup.base_ring()
+            5-adic Field with floating precision 8
+
+        If you decrease the precision, the precision of the base stays the same::
+
+            sage: Kdown = K.change(prec=2); Kdown
+            Unramified Extension in a defined by x^3 + 3*x + 3 with floating precision 2 over 5-adic Field
+            sage: Kdown.precision_cap()
+            2
+            sage: Kdown.base_ring()
+            5-adic Field with floating precision 4
+
+        Changing the prime works for extensions::
+
+            sage: x = polygen(ZZ)
+            sage: R.<a> = Zp(5).extension(x^2 + 2)
+            sage: S = R.change(p=7)
+            sage: S.defining_polynomial(exact=True)
+            x^2 + 2
+            sage: A.<y> = Zp(5)[]
+            sage: R.<a> = Zp(5).extension(y^2 + 2)
+            sage: S = R.change(p=7)
+            sage: S.defining_polynomial(exact=True)
+            y^2 + 2
+
+        ::
+
+            sage: R.<a> = Zq(5^3)
+            sage: S = R.change(prec=50)
+            sage: S.defining_polynomial(exact=True)
+            x^3 + 3*x + 3
+        """
+        # We support both print_* and * for *=mode, pos, sep, alphabet
+        for atr in ('print_mode', 'print_pos', 'print_sep', 'print_alphabet'):
+            if atr in kwds:
+                kwds[atr[6:]] = kwds.pop(atr)
+        def get_unramified_modulus(q, res_name):
+            from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
+            return GF(q, res_name).modulus().change_ring(ZZ)
+        n = None
+        q = None
+        from .padic_base_generic import pAdicBaseGeneric
+        if 'q' in kwds and isinstance(self.base_ring(), pAdicBaseGeneric):
+            q = kwds.pop('q')
+            if not isinstance(q, Integer):
+                raise TypeError("q must be an integer")
+            p, n = q.is_prime_power(get_data=True)
+            if n == 0:
+                raise ValueError("q must be a prime power")
+            if 'p' in kwds and kwds['p'] != p:
+                raise ValueError("q does not match p")
+            kwds['p'] = p
+        functor, ring = self.construction()
+        functor = copy(functor)
+        if 'mode' in kwds and 'show_prec' not in kwds:
+            new_type = kwds.get('type', self._prec_type())
+            cur_type = self._prec_type()
+            cur_mode = self._printer._print_mode()
+            cur_show_prec = self._printer._show_prec()
+            from .factory import _default_show_prec
+            if cur_show_prec == _default_show_prec(cur_type, cur_mode):
+                kwds['show_prec'] = _default_show_prec(new_type, kwds['mode'])
+            else:
+                raise RuntimeError
+        p = kwds.get('p', functor.p if hasattr(functor, 'p') else self.prime())
+        curpstr = str(self.prime())
+        functor_dict = getattr(functor, "extras", getattr(functor, "kwds", None))
+        # If we are switching to 'digits', or changing p, need to ensure a large enough alphabet.
+        if 'alphabet' not in kwds and (kwds.get('mode') == 'digits' or
+           (functor_dict['print_mode'].get('mode') == 'digits' and p > getattr(functor, "p", p))):
+            from .padic_printing import _printer_defaults
+            kwds['alphabet'] = _printer_defaults.alphabet()[:p]
+
+        # There are two kinds of functors possible:
+        # CompletionFunctor and AlgebraicExtensionFunctor
+        # We distinguish them by the presence of ``prec``,
+        if hasattr(functor, "prec"):
+            functor.extras = copy(functor.extras)
+            functor.extras['print_mode'] = copy(functor.extras['print_mode'])
+            if 'type' in kwds and kwds['type'] not in functor._dvr_types:
+                raise ValueError("completion type must be one of %s"%(", ".join(functor._dvr_types[1:])))
+            if 'field' in kwds:
+                field = kwds.pop('field')
+                if field:
+                    ring = ring.fraction_field()
+                    if 'type' not in kwds:
+                        if self._prec_type() == 'capped-abs':
+                            kwds['type'] = 'capped-rel'
+                        elif self._prec_type() == 'fixed-mod':
+                            raise TypeError('You must specify the type explicitly')
+                elif ring.is_field():
+                    ring = ring.ring_of_integers()
+            for atr in ('p', 'prec', 'type'):
+                if atr in kwds:
+                    setattr(functor, atr, kwds.pop(atr))
+            if q is not None:
+                if 'names' in kwds:
+                    names = kwds.pop('names')
+                elif 'unram_name' in kwds:
+                    names = kwds.pop('unram_name')
+                else:
+                    raise TypeError("You must specify the name of the generator")
+                res_name = kwds.pop('res_name', names + '0')
+                modulus = kwds.pop('modulus', get_unramified_modulus(q, res_name))
+                implementation = kwds.pop('implementation', 'FLINT')
+            # We have to change the way p prints in the default case
+            if 'names' in kwds:
+                functor.extras['names'] = kwds.pop('names')
+            elif functor.extras['names'][0] == curpstr:
+                functor.extras['names'] = (str(p),)
+            for atr in ('ram_name', 'var_name'):
+                if atr in kwds:
+                    functor.extras['print_mode'][atr] = kwds.pop(atr)
+                elif functor.extras['print_mode'].get(atr) == curpstr:
+                    functor.extras['print_mode'][atr] = str(p)
+            if 'check' in kwds:
+                functor.extras['check'] = kwds.pop('check')
+            for atr in ('mode', 'pos', 'unram_name', 'max_ram_terms', 'max_unram_terms', 'max_terse_terms', 'sep', 'alphabet', 'show_prec'):
+                if atr in kwds:
+                    functor.extras['print_mode'][atr] = kwds.pop(atr)
+            if kwds:
+                raise ValueError("Extra arguments received: %s"%(", ".join(kwds.keys())))
+            if q is not None:
+                # Create an unramified extension
+                base = functor(ring)
+                from .factory import ExtensionFactory
+                modulus = modulus.change_ring(base)
+                return ExtensionFactory(base=base, premodulus=modulus, names=names, res_name=res_name, unram=True, implementation=implementation)
+        else:
+            functor.kwds = copy(functor.kwds)
+            functor.kwds['print_mode'] = copy(functor.kwds['print_mode'])
+            if 'prec' in kwds:
+                prec = kwds.pop('prec')
+                baseprec = (prec - 1) // self.e() + 1
+                if baseprec > self.base_ring().precision_cap():
+                    kwds['prec'] = baseprec
+                functor.kwds['prec'] = prec
+            from sage.rings.padics.padic_base_generic import pAdicBaseGeneric
+            if 'names' in kwds:
+                functor.names = [kwds.pop('names')]
+            modulus = None
+            if 'modulus' in kwds:
+                modulus = kwds.pop('modulus')
+                if n is not None and modulus.degree() != n:
+                    raise ValueError("modulus must have degree matching q")
+            elif q is not None and self.f() != 1:
+                # If q is specified, replace the modulus with one from q.
+                modulus = get_unramified_modulus(q, functor.kwds.get('res_name', functor.names[0] + '0'))
+            for atr in ('var_name', 'res_name', 'unram_name', 'ram_name'):
+                if atr in kwds:
+                    functor.kwds[atr] = kwds.pop(atr)
+            if 'check' in kwds:
+                functor.kwds['check'] = kwds['check']
+            for atr in ('mode', 'pos', 'max_ram_terms', 'max_unram_terms', 'max_terse_terms', 'sep', 'alphabet', 'show_prec'):
+                if atr in kwds:
+                    functor.kwds['print_mode'][atr] = kwds[atr]
+            if 'base' in kwds:
+                ring = kwds['base']
+            else:
+                if q is not None and self.f() == 1:
+                    kwds['q'] = q
+                ring = ring.change(**kwds)
+            if modulus is None:
+                if len(functor.polys) != 1:
+                    raise RuntimeError("Unexpected number of defining polynomials")
+                modulus = functor.polys[0]
+            if isinstance(modulus.base_ring(), pAdicBaseGeneric):
+                modulus.change_ring(ring)
+            functor.polys = [modulus]
+        return functor(ring)
+
     def precision_cap(self):
         r"""
         Returns the precision cap for ``self``.
@@ -562,6 +836,49 @@ def ext(self, *args, **kwds):
         """
         return self.extension(*args, **kwds)
 
+    def _test_add_bigoh(self, **options):
+        r"""
+        Perform tests on ``add_bigoh``.
+
+        EXAMPLES::
+
+            sage: K = Qp(3)
+            sage: K._test_add_bigoh()
+
+        """
+        tester = self._tester(**options)
+        for x in tester.some_elements():
+            tester.assertEqual(x.add_bigoh(x.precision_absolute()), x)
+            from sage.rings.all import infinity
+            tester.assertEqual(x.add_bigoh(infinity), x)
+            tester.assertEqual(x.add_bigoh(x.precision_absolute()+1), x)
+
+            y = x.add_bigoh(0)
+            tester.assertIs(y.parent(), self)
+            if self.is_capped_absolute():
+                tester.assertEqual(y.precision_absolute(), 0)
+                tester.assertEqual(y, self.zero())
+            elif self.is_capped_relative():
+                tester.assertLessEqual(y.precision_absolute(), 0)
+            elif self.is_fixed_mod() or self.is_floating_point():
+                tester.assertGreaterEqual((x-y).valuation(), 0)
+            else:
+                raise NotImplementedError
+
+            # if absprec < 0, then the result is in the fraction field (see #13591)
+            try:
+                y = x.add_bigoh(-1)
+            except ValueError:
+                tester.assertTrue(self.is_fixed_mod())
+            else:
+                tester.assertIs(y.parent(), self.fraction_field())
+                if not self.is_floating_point():
+                    tester.assertLessEqual(y.precision_absolute(), -1)
+
+            # make sure that we handle very large values correctly
+            absprec = Integer(2)**1000
+            tester.assertEqual(x.add_bigoh(absprec), x)
+
     def _test_residue(self, **options):
         r"""
         Perform some tests on the residue field of this ring.
diff --git a/src/sage/rings/padics/local_generic_element.pyx b/src/sage/rings/padics/local_generic_element.pyx
index f2fb220e4be..85c1d318b15 100644
--- a/src/sage/rings/padics/local_generic_element.pyx
+++ b/src/sage/rings/padics/local_generic_element.pyx
@@ -8,11 +8,12 @@ AUTHORS:
 
 - David Roe: initial version
 
-- Julian Rueth (2012-10-15): added inverse_of_unit()
+- Julian Rueth (2012-10-15, 2014-06-25): added inverse_of_unit(); improved
+  add_bigoh()
 """
 #*****************************************************************************
-#       Copyright (C) 2007,2008,2009 David Roe <roed@math.harvard.edu>
-#                     2012 Julian Rueth <julian.rueth@fsfe.org>
+#       Copyright (C) 2007-2013 David Roe <roed@math.harvard.edu>
+#                     2012-2014 Julian Rueth <julian.rueth@fsfe.org>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
 #  as published by the Free Software Foundation; either version 2 of
@@ -381,21 +382,75 @@ cdef class LocalGenericElement(CommutativeRingElement):
         # this doctest doesn't actually test this function, since _sub_ is overridden.
         return self + (-right)
 
-    def add_bigoh(self, prec):
+    def add_bigoh(self, absprec):
         """
-        Returns self to reduced precision ``prec``.
+        Return a copy of this element with ablsolute precision decreased to
+        ``absprec``.
+
+        INPUT:
+
+        - ``absprec`` -- an integer or positive infinity
 
         EXAMPLES::
-            sage: K = Qp(11, 5)
-            sage: L.<a> = K.extension(x^20 - 11)
-            sage: b = a^3 + 3*a^5; b
-            a^3 + 3*a^5 + O(a^103)
-            sage: b.add_bigoh(17)
-            a^3 + 3*a^5 + O(a^17)
-            sage: b.add_bigoh(150)
-            a^3 + 3*a^5 + O(a^103)
+
+            sage: K = QpCR(3,4)
+            sage: o = K(1); o
+            1 + O(3^4)
+            sage: o.add_bigoh(2)
+            1 + O(3^2)
+            sage: o.add_bigoh(-5)
+            O(3^-5)
+
+        One cannot use ``add_bigoh`` to lift to a higher precision; this
+        can be accomplished with :meth:`lift_to_precision`::
+
+            sage: o.add_bigoh(5)
+            1 + O(3^4)
+
+        Negative values of ``absprec`` return an element in the fraction field
+        of the element's parent::
+
+            sage: R = ZpCA(3,4)
+            sage: R(3).add_bigoh(-5)
+            O(3^-5)
+
+        For fixed-mod elements this method truncates the element::
+
+            sage: R = ZpFM(3,4)
+            sage: R(3).add_bigoh(1)
+            O(3^4)
+
+        If ``absprec`` exceeds the precision of the element, then this method
+        has no effect::
+
+            sage: R(3).add_bigoh(5)
+            3 + O(3^4)
+
+        However, a negative value for ``absprec`` leads to an error, since
+        there is no fraction field for fixed-mod elements::
+
+            sage: R(3).add_bigoh(-1)
+            Traceback (most recent call last):
+            ...
+            ValueError: absprec must be at least 0
+
+        TESTS:
+
+        Test that this also works for infinity::
+
+            sage: R = ZpCR(3,4)
+            sage: R(3).add_bigoh(infinity)
+            3 + O(3^5)
+            sage: R(0).add_bigoh(infinity)
+            0
+
         """
-        return self.parent()(self, absprec=prec)
+        parent = self.parent()
+        if absprec >= self.precision_absolute():
+            return self
+        if absprec < 0:
+            parent = parent.fraction_field()
+        return parent(self, absprec=absprec)
 
     #def copy(self):
     #    raise NotImplementedError
diff --git a/src/sage/rings/padics/misc.py b/src/sage/rings/padics/misc.py
index e78b1efcc73..bdd5b5a14b4 100644
--- a/src/sage/rings/padics/misc.py
+++ b/src/sage/rings/padics/misc.py
@@ -1,16 +1,19 @@
+# -*- coding: utf-8 -*-
 r"""
 Miscellaneous Functions
 
-This file contains two miscellaneous functions used by `p`-adics.
+This file contains several miscellaneous functions used by `p`-adics.
 
+- ``gauss_sum`` -- compute Gauss sums using the Gross-Koblitz formula.
 - ``min`` -- a version of ``min`` that returns `\infty` on empty input.
 - ``max`` -- a version of ``max`` that returns `-\infty` on empty input.
 
 AUTHORS:
 
 - David Roe
+- Adriana Salerno
+- Ander Steele
 """
-
 #*****************************************************************************
 #       Copyright (C) 2007-2013 David Roe <roed.math@gmail.com>
 #                               William Stein <wstein@gmail.com>
@@ -27,6 +30,94 @@
 from six.moves.builtins import max as python_max
 from sage.rings.infinity import infinity
 
+def gauss_sum(a, p, f, prec=20):
+    r"""
+    Return the Gauss sum `g_q(a)` as a `p`-adic number.
+
+    The Gauss sum `g_q(a)` is defined by
+
+    .. MATH::
+
+        g_q(a)= \sum_{u\in F_q^*} \omega(u)^{-a} \zeta_q^u,
+
+    where `q = p^f`, `\omega` is the Teichmüller character and
+    `\zeta_q` is some arbitrary choice of primitive `q`-th root of
+    unity. The computation is adapted from the main theorem in Alain
+    Robert's paper *The Gross-Koblitz formula revisited*,
+    Rend. Sem. Mat. Univ. Padova 105 (2001), 157--170.
+
+    Let `p` be a prime, `f` a positive integer, `q=p^f`, and `\pi` be
+    a root of `f(x) = x^{p-1}+p`.  Let `0\leq a < q-1`. Then the
+    Gross-Koblitz formula gives us the value of the Gauss sum `g_q(a)`
+    as a product of p-adic Gamma functions as follows:
+
+    .. MATH::
+
+        g_q(a) = \pi^s \prod_{0\leq i < f} \Gamma_p(a^{(i)}/(q-1)),
+
+    where `s` is the sum of the digits of `a` in base `p` and the
+    `a^{(i)}` have `p`-adic expansions obtained from cyclic
+    permutations of that of `a`.
+
+    INPUT:
+
+    - ``a`` -- integer
+
+    - ``p`` -- prime
+
+    - ``f`` -- positive integer
+
+    - ``prec`` -- positive integer (optional, 20 by default)
+
+    OUTPUT:
+
+    a `p`-adic number in an Eisenstein extension of `\QQ_p`
+
+    .. NOTE::
+
+        This is based on GP code written by Adriana Salerno.
+
+    EXAMPLES:
+
+    In this example, we verify that `g_3(0) = -1`::
+
+        sage: from sage.rings.padics.misc import gauss_sum
+        sage: -gauss_sum(0,3,1)
+        1 + O(pi^40)
+
+    Next, we verify that `g_5(a) g_5(-a) = 5 (-1)^a`::
+
+        sage: from sage.rings.padics.misc import gauss_sum
+        sage: gauss_sum(2,5,1)^2-5
+        O(pi^84)
+        sage: gauss_sum(1,5,1)*gauss_sum(3,5,1)+5
+        O(pi^84)
+
+    Finally, we compute a non-trivial value::
+
+        sage: from sage.rings.padics.misc import gauss_sum
+        sage: gauss_sum(2,13,2)
+        6*pi^2 + 7*pi^14 + 11*pi^26 + 3*pi^62 + 6*pi^74 + 3*pi^86 + 5*pi^98 +
+        pi^110 + 7*pi^134 + 9*pi^146 + 4*pi^158 + 6*pi^170 + 4*pi^194 +
+        pi^206 + 6*pi^218 + 9*pi^230 + O(pi^242)
+    """
+    from sage.rings.padics.factory import Zp
+    from sage.rings.all import PolynomialRing
+    a = a % (p**f - 1)
+    R = Zp(p, prec)
+    X = PolynomialRing(R, name='X').gen()
+    pi = R.ext(X**(p - 1) + p, names='pi').gen()
+    digits = Zp(p)(a).list(start_val=0)
+    n = len(digits)
+    digits = digits + [0] * (f - n)
+    s = sum(digits)
+    out = -pi**s
+    for i in range(f):
+        a_i = R.sum(digits[k] * p**((i + k) % f) for k in range(f))
+        if a_i:
+            out *= R((a_i / (p**f - 1)).gamma())
+    return out
+
 
 def min(*L):
     r"""
diff --git a/src/sage/rings/padics/padic_ZZ_pX_CR_element.pyx b/src/sage/rings/padics/padic_ZZ_pX_CR_element.pyx
index 28d5a0ed9ee..ba4bdd18a63 100644
--- a/src/sage/rings/padics/padic_ZZ_pX_CR_element.pyx
+++ b/src/sage/rings/padics/padic_ZZ_pX_CR_element.pyx
@@ -859,8 +859,8 @@ cdef class pAdicZZpXCRElement(pAdicZZpXElement):
             '5563A4105291255628.148272'
             sage: repr(y*847)[3:]
             '000000000000000000000003'
-            sage: repr(W(77/3, relprec=0))[3:]
-            '000'
+            sage: repr(W(77/3, relprec=0))
+            '0'
             sage: c = F(11^-1 + O(11^2)); repr(c)[3:]
             '011111.01A'
             sage: repr(c * 11)[3:]
diff --git a/src/sage/rings/padics/padic_extension_generic.py b/src/sage/rings/padics/padic_extension_generic.py
index 6dab5a1e08a..fa317a30abf 100644
--- a/src/sage/rings/padics/padic_extension_generic.py
+++ b/src/sage/rings/padics/padic_extension_generic.py
@@ -321,6 +321,32 @@ def polynomial_ring(self):
     #        xnew = x - x*delta*(1-q*delta)
     #    return x
 
+    def construction(self):
+        """
+        Returns the functorial construction of this ring, namely,
+        the algebraic extension of the base ring defined by the given
+        polynomial.
+
+        Also preserves other information that makes this ring unique
+        (e.g. precision, rounding, print mode).
+
+        EXAMPLES::
+
+            sage: R.<a> = Zq(25, 8, print_mode='val-unit')
+            sage: c, R0 = R.construction(); R0
+            5-adic Ring with capped relative precision 8
+            sage: c(R0)
+            Unramified Extension in a defined by x^2 + 4*x + 2 with capped relative precision 8 over 5-adic Ring
+            sage: c(R0) == R
+            True
+        """
+        from sage.categories.pushout import AlgebraicExtensionFunctor as AEF
+        print_mode = self._printer.dict()
+        return (AEF([self.defining_polynomial(exact=True)], [self.variable_name()],
+                    prec=self.precision_cap(), print_mode=self._printer.dict(),
+                    implementation=self._implementation),
+                self.base_ring())
+
     def fraction_field(self, print_mode=None):
         r"""
         Returns the fraction field of this extension, which is just
@@ -346,18 +372,10 @@ def fraction_field(self, print_mode=None):
         """
         if self.is_field() and print_mode is None:
             return self
-        print_mode = self._modified_print_mode(print_mode)
-        ground_mode = print_mode.copy()
-        # We don't want to confuse the ground ring with different names.
-        ground_mode['ram_name'] = None
-        ground_mode['unram_name'] = None
-        K = self.ground_ring().fraction_field(ground_mode)
-        #we don't want to set the print options due to the ground ring since
-        #different extension fields (with different options) can share the same ground ring.
-        if self.is_lazy():
-            return K.extension(self._exact_modulus, prec = self.precision_cap(), halt = self.halting_parameter(), res_name = self.residue_field().variable_name(), print_mode=print_mode, implementation=self._implementation)
+        if print_mode is None:
+            return self.change(field=True)
         else:
-            return K.extension(self._exact_modulus, prec = self.precision_cap(), res_name = self.residue_field().variable_name(), print_mode=print_mode, implementation=self._implementation)
+            return self.change(field=True, **print_mode)
 
     def integer_ring(self, print_mode=None):
         r"""
@@ -385,18 +403,10 @@ def integer_ring(self, print_mode=None):
         #Currently does not support fields with non integral defining polynomials.  This should change when the padic_general_extension framework gets worked out.
         if not self.is_field() and print_mode is None:
             return self
-        print_mode = self._modified_print_mode(print_mode)
-        ground_mode = print_mode.copy()
-        # We don't want to confuse the ground ring with different names.
-        ground_mode['ram_name'] = None
-        ground_mode['unram_name'] = None
-        K = self.ground_ring().integer_ring(ground_mode)
-        #we don't want to set the print options due to the ground ring since
-        #different extension fields (with different options) can share the same ground ring.
-        if self.is_lazy():
-            return K.extension(self._exact_modulus, prec = self.precision_cap(), halt = self.halting_parameter(), res_name = self.residue_field().variable_name(), print_mode=print_mode)
+        if print_mode is None:
+            return self.change(field=False)
         else:
-            return K.extension(self._exact_modulus, prec = self.precision_cap(), res_name = self.residue_field().variable_name(), print_mode=print_mode)
+            return self.change(field=False, **print_mode)
 
     #def hasGNB(self):
     #    raise NotImplementedError
diff --git a/src/sage/rings/padics/padic_generic.py b/src/sage/rings/padics/padic_generic.py
index 466a3d402b9..c43c74e6aa5 100644
--- a/src/sage/rings/padics/padic_generic.py
+++ b/src/sage/rings/padics/padic_generic.py
@@ -108,7 +108,7 @@ def _modified_print_mode(self, print_mode):
             print_mode = {}
         elif isinstance(print_mode, str):
             print_mode = {'mode': print_mode}
-        for option in ['mode', 'pos', 'ram_name', 'unram_name', 'var_name', 'max_ram_terms', 'max_unram_terms', 'max_terse_terms', 'sep', 'alphabet']:
+        for option in ['mode', 'pos', 'ram_name', 'unram_name', 'var_name', 'max_ram_terms', 'max_unram_terms', 'max_terse_terms', 'sep', 'alphabet', 'show_prec']:
             if option not in print_mode:
                 print_mode[option] = self._printer.dict()[option]
         return print_mode
diff --git a/src/sage/rings/padics/padic_generic_element.pyx b/src/sage/rings/padics/padic_generic_element.pyx
index f6d58364616..ceff1efe618 100644
--- a/src/sage/rings/padics/padic_generic_element.pyx
+++ b/src/sage/rings/padics/padic_generic_element.pyx
@@ -1266,6 +1266,61 @@ cdef class pAdicGenericElement(LocalGenericElement):
             #won't work for general extensions...
             return (self.valuation() % 2 == 0) and (self.unit_part().residue(3) == 1)
 
+    def is_squarefree(self):
+        r"""
+        Return whether this element is squarefree, i.e., whether there exists
+        no non-unit `g` such that `g^2` divides this element.
+
+        EXAMPLES:
+
+        The zero element is never squarefree::
+
+            sage: K = Qp(2)
+            sage: K.zero().is_squarefree()
+            False
+
+        In `p`-adic rings, only elements of valuation at most 1 are
+        squarefree::
+
+            sage: R = Zp(2)
+            sage: R(1).is_squarefree()
+            True
+            sage: R(2).is_squarefree()
+            True
+            sage: R(4).is_squarefree()
+            False
+
+        This works only if the precision is known sufficiently well::
+
+            sage: R(0,1).is_squarefree()
+            Traceback (most recent call last):
+            ...
+            PrecisionError: element not known to sufficient precision to decide squarefreeness
+            sage: R(0,2).is_squarefree()
+            False
+            sage: R(1,1).is_squarefree()
+            True
+
+        For fields we are not so strict about the precision and treat inexact
+        zeros as the zero element::
+
+            K(0,0).is_squarefree()
+            False
+
+        """
+        if self.parent().is_field():
+            if self.is_zero():
+                return False
+            return True
+        else:
+            v = self.valuation()
+            if v >= 2:
+                return False
+            elif self.is_zero():
+                raise PrecisionError("element not known to sufficient precision to decide squarefreeness")
+            else:
+                return True
+
     #def log_artin_hasse(self):
     #    raise NotImplementedError
 
diff --git a/src/sage/rings/padics/pow_computer.pxd b/src/sage/rings/padics/pow_computer.pxd
index 7d754e8a28c..cce87bb7619 100644
--- a/src/sage/rings/padics/pow_computer.pxd
+++ b/src/sage/rings/padics/pow_computer.pxd
@@ -19,7 +19,7 @@ cdef class PowComputer_class(SageObject):
 
     cdef Integer pow_Integer(self, long n)
     cdef mpz_srcptr pow_mpz_t_top(self)
-    cdef mpz_srcptr pow_mpz_t_tmp(self, unsigned long n)
+    cdef mpz_srcptr pow_mpz_t_tmp(self, long n) except NULL
     cdef mpz_t temp_m
 
 cdef class PowComputer_base(PowComputer_class):
diff --git a/src/sage/rings/padics/pow_computer.pyx b/src/sage/rings/padics/pow_computer.pyx
index 5c6073ee0e4..b56c7bd9b02 100644
--- a/src/sage/rings/padics/pow_computer.pyx
+++ b/src/sage/rings/padics/pow_computer.pyx
@@ -31,6 +31,7 @@ AUTHORS:
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 import weakref
 from cysignals.memory cimport sig_malloc, sig_free
@@ -189,7 +190,7 @@ cdef class PowComputer_class(SageObject):
                 raise ValueError("result too big")
             return self.pow_Integer(mpz_get_ui(_n.value))
 
-    cdef mpz_srcptr pow_mpz_t_tmp(self, unsigned long n):
+    cdef mpz_srcptr pow_mpz_t_tmp(self, long n) except NULL:
         """
         Provides fast access to an ``mpz_srcptr`` pointing to self.prime^n.
 
@@ -271,7 +272,7 @@ cdef class PowComputer_class(SageObject):
         """
         cdef Integer _n = Integer(n)
         cdef Integer ans = PY_NEW(Integer)
-        mpz_set(ans.value, self.pow_mpz_t_tmp(mpz_get_ui(_n.value)))
+        mpz_set(ans.value, self.pow_mpz_t_tmp(mpz_get_si(_n.value)))
         return ans
 
     cdef mpz_srcptr pow_mpz_t_top(self):
@@ -481,7 +482,9 @@ cdef class PowComputer_base(PowComputer_class):
             mpz_set(self.small_powers[1], prime.value)
         for i in range(2, cache_limit + 1):
             mpz_mul(self.small_powers[i], self.small_powers[i - 1], prime.value)
+        sig_on()
         mpz_pow_ui(self.top_power, prime.value, prec_cap)
+        sig_off()
         self.deg = 1
         self.e = 1
         self.f = 1
@@ -532,7 +535,7 @@ cdef class PowComputer_base(PowComputer_class):
         """
         return self.top_power
 
-    cdef mpz_srcptr pow_mpz_t_tmp(self, unsigned long n):
+    cdef mpz_srcptr pow_mpz_t_tmp(self, long n) except NULL:
         """
         Computes self.prime^n.
 
@@ -541,12 +544,27 @@ cdef class PowComputer_base(PowComputer_class):
             sage: PC = PowComputer(3, 5, 10)
             sage: PC._pow_mpz_t_tmp_test(4)
             81
+            sage: PC._pow_mpz_t_tmp_test(-1)
+            Traceback (most recent call last):
+            ...
+            ValueError: n must be non-negative
+
         """
+        if n < 0:
+            raise ValueError("n must be non-negative")
         if n <= self.cache_limit:
             return self.small_powers[n]
         if n == self.prec_cap:
             return self.top_power
+        # n may exceed self.prec_cap. Very large values can, however, lead to
+        # out-of-memory situations in the following computation. This
+        # sig_on()/sig_off() prevents sage from crashing in such cases.
+        # It does not have a significant impact on performance. For small
+        # values of n the powers are taken from self.small_powers, for large
+        # values, the computation dominates the cost of the sig_on()/sig_off().
+        sig_on()
         mpz_pow_ui(self.temp_m, self.prime.value, n)
+        sig_off()
         return self.temp_m
 
 pow_comp_cache = {}
@@ -573,13 +591,13 @@ cdef PowComputer_base PowComputer_c(Integer m, Integer cache_limit, Integer prec
         if PC is not None:
             return PC
     if prec_type == 'capped-rel':
-        from padic_capped_relative_element import PowComputer_ as PC_class
+        from .padic_capped_relative_element import PowComputer_ as PC_class
     elif prec_type == 'capped-abs':
-        from padic_capped_absolute_element import PowComputer_ as PC_class
+        from .padic_capped_absolute_element import PowComputer_ as PC_class
     elif prec_type == 'fixed-mod':
-        from padic_fixed_mod_element import PowComputer_ as PC_class
+        from .padic_fixed_mod_element import PowComputer_ as PC_class
     elif prec_type == 'floating-point':
-        from padic_floating_point_element import PowComputer_ as PC_class
+        from .padic_floating_point_element import PowComputer_ as PC_class
     else:
         PC_class = PowComputer_base
     PC = PC_class(m, mpz_get_ui(cache_limit.value), mpz_get_ui(prec_cap.value), mpz_get_ui(prec_cap.value), in_field)
diff --git a/src/sage/rings/padics/pow_computer_ext.pxd b/src/sage/rings/padics/pow_computer_ext.pxd
index 7bb16e2811b..e87f801971f 100644
--- a/src/sage/rings/padics/pow_computer_ext.pxd
+++ b/src/sage/rings/padics/pow_computer_ext.pxd
@@ -15,7 +15,7 @@ cdef class PowComputer_ext(PowComputer_class):
     cdef object _shift_seed
     cdef object _ext_type
 
-    cdef ZZ_c* pow_ZZ_tmp(self, long n)
+    cdef ZZ_c* pow_ZZ_tmp(self, long n) except NULL
     cdef ZZ_c* pow_ZZ_top(self)
 
     cdef void cleanup_ext(self)
diff --git a/src/sage/rings/padics/pow_computer_ext.pyx b/src/sage/rings/padics/pow_computer_ext.pyx
index 413e89ab369..83ae6c127fc 100644
--- a/src/sage/rings/padics/pow_computer_ext.pyx
+++ b/src/sage/rings/padics/pow_computer_ext.pyx
@@ -47,6 +47,7 @@ AUTHORS:
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 from cpython.list cimport *
 from cpython.dict cimport *
@@ -584,7 +585,7 @@ cdef class PowComputer_ext(PowComputer_class):
         mpz_clear(self.temp_m)
         mpz_clear(self.temp_m2)
 
-    cdef mpz_srcptr pow_mpz_t_tmp(self, unsigned long n):
+    cdef mpz_srcptr pow_mpz_t_tmp(self, long n) except NULL:
         """
         Provides fast access to an mpz_t* pointing to self.prime^n.
 
@@ -607,19 +608,25 @@ cdef class PowComputer_ext(PowComputer_class):
             sage: PC._pow_mpz_t_tmp_test(4) #indirect doctest
             625
         """
-        # READ THE DOCSTRING
         if n < 0:
-            # Exception will be ignored by Cython
-            raise ValueError("n must be positive")
+            raise ValueError("n must be non-negative")
         if n <= self.cache_limit:
             ZZ_to_mpz(self.temp_m, &(self.small_powers[n]))
         elif n == self.prec_cap:
             ZZ_to_mpz(self.temp_m, &self.top_power)
         else:
+            sig_on()
+            # n may exceed self.prec_cap. Very large values can, however, lead to
+            # out-of-memory situations in the following computation. This
+            # sig_on()/sig_off() prevents sage from crashing in such cases.
+            # It does not have a significant impact on performance. For small
+            # values of n the powers are taken from self.small_powers, for large
+            # values, the computation dominates the cost of the sig_on()/sig_off().
             mpz_pow_ui(self.temp_m, self.prime.value, n)
+            sig_off()
         return self.temp_m
 
-    cdef ZZ_c* pow_ZZ_tmp(self, long n):
+    cdef ZZ_c* pow_ZZ_tmp(self, long n) except NULL:
         """
         Provides fast access to a ZZ_c* pointing to self.prime^n.
 
@@ -639,8 +646,7 @@ cdef class PowComputer_ext(PowComputer_class):
             625
         """
         if n < 0:
-            # Exception will be ignored by Cython
-            raise ValueError("n must be positive")
+            raise ValueError("n must be non-negative")
         if n <= self.cache_limit:
             return &(self.small_powers[n])
         if n == self.prec_cap:
diff --git a/src/sage/rings/padics/pow_computer_flint.pyx b/src/sage/rings/padics/pow_computer_flint.pyx
index d5542626d5c..79d2f339fb1 100644
--- a/src/sage/rings/padics/pow_computer_flint.pyx
+++ b/src/sage/rings/padics/pow_computer_flint.pyx
@@ -134,7 +134,7 @@ cdef class PowComputer_flint(PowComputer_class):
             fmpz_pow_ui(self._fpow_variable, self.fprime, n)
             return &self._fpow_variable
 
-    cdef mpz_srcptr pow_mpz_t_tmp(self, unsigned long n):
+    cdef mpz_srcptr pow_mpz_t_tmp(self, long n) except NULL:
         """
         Returns a pointer to an ``mpz_t`` holding `p^n`.
 
diff --git a/src/sage/rings/padics/qadic_flint_CA.pyx b/src/sage/rings/padics/qadic_flint_CA.pyx
index 745fd3233c6..fec455a14cf 100644
--- a/src/sage/rings/padics/qadic_flint_CA.pyx
+++ b/src/sage/rings/padics/qadic_flint_CA.pyx
@@ -1,5 +1,3 @@
-from types import MethodType
-
 include "sage/libs/linkages/padics/fmpz_poly_unram.pxi"
 include "sage/libs/linkages/padics/unram_shared.pxi"
 include "CA_template.pxi"
@@ -24,9 +22,9 @@ cdef class PowComputer_(PowComputer_flint_unram):
         PowComputer_flint_unram.__init__(self, prime, cache_limit, prec_cap, ram_prec_cap, in_field, poly)
 
 cdef class qAdicCappedAbsoluteElement(CAElement):
-    frobenius = MethodType(frobenius_unram, None, qAdicCappedAbsoluteElement)
-    trace = MethodType(trace_unram, None, qAdicCappedAbsoluteElement)
-    norm = MethodType(norm_unram, None, qAdicCappedAbsoluteElement)
+    frobenius = frobenius_unram
+    trace = trace_unram
+    norm = norm_unram
 
     def matrix_mod_pn(self):
         """
diff --git a/src/sage/rings/padics/qadic_flint_CR.pyx b/src/sage/rings/padics/qadic_flint_CR.pyx
index 988316ab748..0c48a5a7ec9 100644
--- a/src/sage/rings/padics/qadic_flint_CR.pyx
+++ b/src/sage/rings/padics/qadic_flint_CR.pyx
@@ -1,5 +1,3 @@
-from types import MethodType
-
 include "sage/libs/linkages/padics/fmpz_poly_unram.pxi"
 include "sage/libs/linkages/padics/unram_shared.pxi"
 include "CR_template.pxi"
@@ -24,9 +22,9 @@ cdef class PowComputer_(PowComputer_flint_unram):
         PowComputer_flint_unram.__init__(self, prime, cache_limit, prec_cap, ram_prec_cap, in_field, poly)
 
 cdef class qAdicCappedRelativeElement(CRElement):
-    frobenius = MethodType(frobenius_unram, None, qAdicCappedRelativeElement)
-    trace = MethodType(trace_unram, None, qAdicCappedRelativeElement)
-    norm = MethodType(norm_unram, None, qAdicCappedRelativeElement)
+    frobenius = frobenius_unram
+    trace = trace_unram
+    norm = norm_unram
 
     def matrix_mod_pn(self):
         """
diff --git a/src/sage/rings/padics/qadic_flint_FM.pyx b/src/sage/rings/padics/qadic_flint_FM.pyx
index 0075726647e..f8210a8d1ed 100644
--- a/src/sage/rings/padics/qadic_flint_FM.pyx
+++ b/src/sage/rings/padics/qadic_flint_FM.pyx
@@ -1,5 +1,3 @@
-from types import MethodType
-
 include "sage/libs/linkages/padics/fmpz_poly_unram.pxi"
 include "sage/libs/linkages/padics/unram_shared.pxi"
 include "FM_template.pxi"
@@ -24,9 +22,9 @@ cdef class PowComputer_(PowComputer_flint_unram):
         PowComputer_flint_unram.__init__(self, prime, cache_limit, prec_cap, ram_prec_cap, in_field, poly)
 
 cdef class qAdicFixedModElement(FMElement):
-    frobenius = MethodType(frobenius_unram, None, qAdicFixedModElement)
-    trace = MethodType(trace_unram, None, qAdicFixedModElement)
-    norm = MethodType(norm_unram, None, qAdicFixedModElement)
+    frobenius = frobenius_unram
+    trace = trace_unram
+    norm = norm_unram
 
     def matrix_mod_pn(self):
         """
diff --git a/src/sage/rings/padics/qadic_flint_FP.pyx b/src/sage/rings/padics/qadic_flint_FP.pyx
index 0137a9f2da6..3f6b66568e3 100644
--- a/src/sage/rings/padics/qadic_flint_FP.pyx
+++ b/src/sage/rings/padics/qadic_flint_FP.pyx
@@ -1,5 +1,3 @@
-from types import MethodType
-
 include "sage/libs/linkages/padics/fmpz_poly_unram.pxi"
 include "sage/libs/linkages/padics/unram_shared.pxi"
 include "FP_template.pxi"
@@ -24,9 +22,9 @@ cdef class PowComputer_(PowComputer_flint_unram):
         PowComputer_flint_unram.__init__(self, prime, cache_limit, prec_cap, ram_prec_cap, in_field, poly)
 
 cdef class qAdicFloatingPointElement(FPElement):
-    frobenius = MethodType(frobenius_unram, None, qAdicFloatingPointElement)
-    trace = MethodType(trace_unram, None, qAdicFloatingPointElement)
-    norm = MethodType(norm_unram, None, qAdicFloatingPointElement)
+    frobenius = frobenius_unram
+    trace = trace_unram
+    norm = norm_unram
 
     def matrix_mod_pn(self):
         """
diff --git a/src/sage/rings/polynomial/laurent_polynomial.pxd b/src/sage/rings/polynomial/laurent_polynomial.pxd
index 6f6d2dc6669..d78dae43741 100644
--- a/src/sage/rings/polynomial/laurent_polynomial.pxd
+++ b/src/sage/rings/polynomial/laurent_polynomial.pxd
@@ -1,21 +1,22 @@
-from sage.rings.quotient_ring_element import QuotientRingElement
 from sage.structure.element cimport CommutativeAlgebraElement, ModuleElement, RingElement, Element
 from sage.rings.polynomial.polydict cimport ETuple, PolyDict
 from sage.rings.polynomial.multi_polynomial cimport MPolynomial
-from sage.rings.polynomial.polynomial_element cimport Polynomial
 
-cdef class LaurentPolynomial_generic(CommutativeAlgebraElement):
-    pass
 
-cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
+cdef class LaurentPolynomial(CommutativeAlgebraElement):
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
+    cpdef _floordiv_(self, other)
+    cpdef long number_of_terms(self) except -1
+
+
+cdef class LaurentPolynomial_univariate(LaurentPolynomial):
     cpdef ModuleElement __u
     cdef long __n
-    cpdef long number_of_terms(self)
 
-cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
+
+cdef class LaurentPolynomial_mpair(LaurentPolynomial):
     cdef ETuple _mon
     cdef MPolynomial _poly
     cdef PolyDict _prod
     cdef _new_c(self)
-    cpdef long number_of_terms(self)
-
diff --git a/src/sage/rings/polynomial/laurent_polynomial.pyx b/src/sage/rings/polynomial/laurent_polynomial.pyx
index b078b2c6d31..48a787bf7e2 100644
--- a/src/sage/rings/polynomial/laurent_polynomial.pyx
+++ b/src/sage/rings/polynomial/laurent_polynomial.pyx
@@ -20,10 +20,54 @@ from sage.rings.polynomial.polynomial_element import Polynomial
 from sage.structure.richcmp cimport richcmp, rich_to_bool
 
 
-cdef class LaurentPolynomial_generic(CommutativeAlgebraElement):
+cdef class LaurentPolynomial(CommutativeAlgebraElement):
     """
-    A generic Laurent polynomial.
+    Base class for Laurent polynomials.
     """
+    cpdef _add_(self, other):
+        """
+        Abstract addition method
+
+        EXAMPLES::
+
+            sage: R.<x> = LaurentPolynomialRing(ZZ)
+            sage: from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
+            sage: LaurentPolynomial._add_(x, x)
+            Traceback (most recent call last):
+            ...
+            NotImplementedError
+        """
+        raise NotImplementedError
+
+    cpdef _mul_(self, other):
+        """
+        Abstract multiplication method
+
+        EXAMPLES::
+
+            sage: R.<x> = LaurentPolynomialRing(ZZ)
+            sage: from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
+            sage: LaurentPolynomial._mul_(x, x)
+            Traceback (most recent call last):
+            ...
+            NotImplementedError
+        """
+        raise NotImplementedError
+
+    cpdef _floordiv_(self, other):
+        """
+        Abstract floor division method
+
+        EXAMPLES::
+
+            sage: R.<x> = LaurentPolynomialRing(ZZ)
+            sage: from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
+            sage: LaurentPolynomial._floordiv_(x, x)
+            Traceback (most recent call last):
+            ...
+            NotImplementedError
+        """
+        raise NotImplementedError
 
     def _integer_(self, ZZ):
         r"""
@@ -128,10 +172,39 @@ cdef class LaurentPolynomial_generic(CommutativeAlgebraElement):
             sage: a.change_ring(GF(3))
             -x^2 + x^-1
         """
-        return self.parent().change_ring(R)(self)
+        return self._parent.change_ring(R)(self)
+
+    cpdef long number_of_terms(self) except -1:
+        """
+        Abstract method for number of terms
+
+        EXAMPLES::
+
+            sage: R.<x> = LaurentPolynomialRing(ZZ)
+            sage: from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial
+            sage: LaurentPolynomial.number_of_terms(x)
+            Traceback (most recent call last):
+            ...
+            NotImplementedError
+        """
+        raise NotImplementedError
+
+    def hamming_weight(self):
+        """
+        Return the number of non-zero coefficients of self. Also called
+        weight, hamming weight or sparsity.
+
+        EXAMPLES::
+
+            sage: R.<x> = LaurentPolynomialRing(ZZ)
+            sage: f = x^3 - 1
+            sage: f.number_of_terms()
+            2
+        """
+        return self.number_of_terms()
 
 
-cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
+cdef class LaurentPolynomial_univariate(LaurentPolynomial):
     """
     A univariate Laurent polynomial in the form of `t^n \cdot f`
     where `f` is a polynomial in `t`.
@@ -518,7 +591,7 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
 
         return self.__u[i - self.__n]
 
-    cpdef long number_of_terms(self):
+    cpdef long number_of_terms(self) except -1:
         """
         Return the number of non-zero coefficients of self. Also called weight,
         hamming weight or sparsity.
@@ -542,8 +615,6 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
         """
         return self.__u.number_of_terms()
 
-    hamming_weight = number_of_terms
-
     def __iter__(self):
         """
         Iterate through the coefficients from the first nonzero one to the
@@ -1509,7 +1580,8 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
         """
         return self[0]
 
-cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
+
+cdef class LaurentPolynomial_mpair(LaurentPolynomial):
     """
     Multivariate Laurent polynomials.
     """
@@ -1817,7 +1889,7 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
         return self._prod.latex(self.parent().latex_variable_names(),
                                 atomic_coefficients=atomic, sortkey=key)
 
-    cpdef long number_of_terms(self):
+    cpdef long number_of_terms(self) except -1:
         """
         Return the number of non-zero coefficients of self. Also called weight,
         hamming weight or sparsity.
@@ -1841,8 +1913,6 @@ cdef class LaurentPolynomial_mpair(LaurentPolynomial_generic):
         """
         return self._poly.number_of_terms()
 
-    hamming_weight = number_of_terms
-
     def __invert__(LaurentPolynomial_mpair self):
         """
         Return the inverse of ``self``.
diff --git a/src/sage/rings/polynomial/multi_polynomial.pxd b/src/sage/rings/polynomial/multi_polynomial.pxd
index 7e101fa3aa3..f19b8e47d46 100644
--- a/src/sage/rings/polynomial/multi_polynomial.pxd
+++ b/src/sage/rings/polynomial/multi_polynomial.pxd
@@ -2,3 +2,4 @@ from sage.structure.element cimport CommutativeRingElement
 
 cdef class MPolynomial(CommutativeRingElement):
     cdef long _hash_c(self) except -1
+    cpdef _mod_(self, right)
diff --git a/src/sage/rings/polynomial/multi_polynomial.pyx b/src/sage/rings/polynomial/multi_polynomial.pyx
index 52b4dcc6c27..ffadee75ed1 100644
--- a/src/sage/rings/polynomial/multi_polynomial.pyx
+++ b/src/sage/rings/polynomial/multi_polynomial.pyx
@@ -1236,7 +1236,7 @@ cdef class MPolynomial(CommutativeRingElement):
             sage: f.sylvester_matrix(g, x).determinant() == f.resultant(g, x)
             True
 
-        TEST:
+        TESTS:
 
         The variable is optional::
 
@@ -2214,7 +2214,7 @@ cdef class MPolynomial(CommutativeRingElement):
                 a += b
                 d = (t-(L[j].real()))/((t-(L[j])) * (t-(L[j].conjugate())) + u**2)
                 c += d
-            #Newton's Method, to to find solutions. Error bound is while less than diameter of our z
+            #Newton's Method, to find solutions. Error bound is while less than diameter of our z
             err = z.diameter()
             zz = z.diameter()
             n = F.degree()
diff --git a/src/sage/rings/polynomial/multi_polynomial_ideal.py b/src/sage/rings/polynomial/multi_polynomial_ideal.py
index 21f60c0b7d3..1ff1017d87b 100644
--- a/src/sage/rings/polynomial/multi_polynomial_ideal.py
+++ b/src/sage/rings/polynomial/multi_polynomial_ideal.py
@@ -2033,7 +2033,7 @@ def quotient(self, J):
             sage: x * (y*z + x) in I
             True
 
-        TEST:
+        TESTS:
 
         This example checks :trac:`16301`::
 
@@ -3110,7 +3110,7 @@ def __lt__(self, other):
             sage: I < J
             True
 
-        TEST:
+        TESTS:
 
             We test to make sure that pickling works with the cached Groebner basis::
 
@@ -3303,7 +3303,7 @@ def __eq__(self, other):
             sage: I == J
             True
 
-        TEST:
+        TESTS:
 
         This example checks :trac:`12802`::
 
diff --git a/src/sage/rings/polynomial/multi_polynomial_libsingular.pxd b/src/sage/rings/polynomial/multi_polynomial_libsingular.pxd
index c4c3576a1dc..730db34658c 100644
--- a/src/sage/rings/polynomial/multi_polynomial_libsingular.pxd
+++ b/src/sage/rings/polynomial/multi_polynomial_libsingular.pxd
@@ -8,6 +8,9 @@ cdef class MPolynomialRing_libsingular
 cdef class MPolynomial_libsingular(MPolynomial):
     cdef poly *_poly
     cdef ring *_parent_ring
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
+    cpdef _floordiv_(self, right)
     cpdef _repr_short_(self)
     cpdef is_constant(self)
     cpdef _homogenize(self, int var)
diff --git a/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx b/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
index 516629d20b7..917289ad1aa 100644
--- a/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
+++ b/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
@@ -353,7 +353,7 @@ cdef class MPolynomialRing_libsingular(MPolynomialRing_generic):
             sage: P(2^32-1)
             4294967295
 
-        TEST:
+        TESTS:
 
         Make sure that a faster coercion map from the base ring is used;
         see :trac:`9944`::
diff --git a/src/sage/rings/polynomial/pbori.pxd b/src/sage/rings/polynomial/pbori.pxd
index 89e9217c604..74072628460 100644
--- a/src/sage/rings/polynomial/pbori.pxd
+++ b/src/sage/rings/polynomial/pbori.pxd
@@ -18,6 +18,8 @@ cdef class BooleanPolynomialRing(MPolynomialRing_generic):
 
 cdef class BooleanPolynomial(MPolynomial):
     cdef PBPoly _pbpoly
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
 
 cdef class BooleSet:
     cdef BooleanPolynomialRing _ring
@@ -30,6 +32,7 @@ cdef class CCuddNavigator:
 cdef class BooleanMonomial(MonoidElement):
     cdef PBMonom _pbmonom
     cdef BooleanPolynomialRing _ring
+    cpdef _mul_(self, other)
 
 cdef class BooleanMonomialVariableIterator:
     cdef object parent
diff --git a/src/sage/rings/polynomial/pbori.pyx b/src/sage/rings/polynomial/pbori.pyx
index ff42c9aaae1..232bec764a1 100644
--- a/src/sage/rings/polynomial/pbori.pyx
+++ b/src/sage/rings/polynomial/pbori.pyx
@@ -181,7 +181,7 @@ REFERENCES:
   available at
   http://www.itwm.fraunhofer.de/fileadmin/ITWM-Media/Zentral/Pdf/Berichte_ITWM/2007/bericht122.pdf
 """
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from cpython.object cimport Py_EQ, Py_NE
 from cysignals.memory cimport sig_malloc, sig_free
@@ -1138,7 +1138,7 @@ cdef class BooleanPolynomialRing(MPolynomialRing_generic):
         -  ``choose_degree`` - choose degree of monomials
            randomly first, rather than monomials uniformly random
 
-        -  ``vars_set`` - list of integer indicies of
+        -  ``vars_set`` - list of integer indices of
            generators of self to use in the generated polynomial
 
 
@@ -1246,7 +1246,7 @@ cdef class BooleanPolynomialRing(MPolynomialRing_generic):
         -  ``monom_counts`` - a list containing total number
            of monomials up to given degree
 
-        -  ``vars_set`` - list of variable indicies to use in
+        -  ``vars_set`` - list of variable indices to use in
            the generated polynomial
 
         -  ``dfirst`` - if ``True`` choose degree
@@ -1289,7 +1289,7 @@ cdef class BooleanPolynomialRing(MPolynomialRing_generic):
         -  ``monom_counts`` - list of number of monomials up
            to given degree
 
-        -  ``vars_set`` - list of variable indicies to use in
+        -  ``vars_set`` - list of variable indices to use in
            the generated monomial
 
 
@@ -1330,7 +1330,7 @@ cdef class BooleanPolynomialRing(MPolynomialRing_generic):
 
         -  ``degree`` - maximum degree
 
-        -  ``vars_set`` - list of variable indicies of self
+        -  ``vars_set`` - list of variable indices of self
 
 
         EXAMPLES::
@@ -2305,7 +2305,7 @@ cdef class BooleanMonomial(MonoidElement):
 
         INPUT:
 
-        -  ``d`` - dictionary with integer indicies
+        - ``d`` -- dictionary with integer indices
 
         EXAMPLES::
 
@@ -4745,7 +4745,7 @@ cdef class PolynomialConstruct:
     def lead(self, x):
         """
         Return the leading monomial of boolean polynomial ``x``, with
-        respect to to the order of parent ring.
+        respect to the order of parent ring.
 
         EXAMPLES::
 
@@ -8039,7 +8039,7 @@ cdef class BooleConstant:
 
     def has_constant_part(self):
         """
-        This is true for for `BooleConstant(1)`.
+        This is true for `BooleConstant(1)`.
 
         EXAMPLES::
 
@@ -8151,7 +8151,7 @@ cdef class MonomialFactory:
     def __init__(self, ring=None):
         """
         Initialized a polynomial factory of ring is given.
-        Otherwise it it initializes a plain constructor.
+        Otherwise it initializes a plain constructor.
 
         EXAMPLES::
 
@@ -8235,7 +8235,7 @@ cdef class PolynomialFactory:
     def lead(self, x):
         """
         Return the leading monomial of boolean polynomial ``x``, with
-        respect to to the order of parent ring.
+        respect to the order of parent ring.
 
         EXAMPLES::
 
diff --git a/src/sage/rings/polynomial/plural.pxd b/src/sage/rings/polynomial/plural.pxd
index 0f0b65991f2..41db2a144e7 100644
--- a/src/sage/rings/polynomial/plural.pxd
+++ b/src/sage/rings/polynomial/plural.pxd
@@ -32,6 +32,8 @@ cdef class ExteriorAlgebra_plural(NCPolynomialRing_plural):
 
 cdef class NCPolynomial_plural(RingElement):
     cdef poly *_poly
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cpdef _repr_short_(self)
     cdef long _hash_c(self)
     cpdef is_constant(self)
diff --git a/src/sage/rings/polynomial/polydict.pyx b/src/sage/rings/polynomial/polydict.pyx
index 0ffb14b6097..d74a5ac04a6 100644
--- a/src/sage/rings/polynomial/polydict.pyx
+++ b/src/sage/rings/polynomial/polydict.pyx
@@ -36,7 +36,7 @@ AUTHORS:
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from libc.string cimport memcpy
 from cpython.dict cimport *
diff --git a/src/sage/rings/polynomial/polynomial_element.pxd b/src/sage/rings/polynomial/polynomial_element.pxd
index a30609da93e..3abb04f8316 100644
--- a/src/sage/rings/polynomial/polynomial_element.pxd
+++ b/src/sage/rings/polynomial/polynomial_element.pxd
@@ -21,6 +21,9 @@ cdef class Polynomial(CommutativeAlgebraElement):
     cpdef bint is_zero(self)
     cpdef bint is_one(self)
 
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
+    cpdef _floordiv_(self, right)
     cpdef Polynomial _mul_trunc_(self, Polynomial right, long n)
     cpdef Polynomial _power_trunc(self, unsigned long n, long prec)
 
diff --git a/src/sage/rings/polynomial/polynomial_element.pyx b/src/sage/rings/polynomial/polynomial_element.pyx
index 5d5c0831549..4cb38958a0a 100644
--- a/src/sage/rings/polynomial/polynomial_element.pyx
+++ b/src/sage/rings/polynomial/polynomial_element.pyx
@@ -52,6 +52,7 @@ from cpython.number cimport PyNumber_TrueDivide, PyNumber_Check
 
 import operator, copy, re
 
+from sage.cpython.wrapperdescr cimport wrapperdescr_fastcall
 import sage.rings.rational
 import sage.rings.integer
 from . import polynomial_ring
@@ -66,7 +67,6 @@ from sage.misc.abstract_method import abstract_method
 from sage.misc.latex import latex
 from sage.misc.long cimport pyobject_to_long
 from sage.structure.factorization import Factorization
-from sage.structure.element import coerce_binop
 from sage.structure.richcmp cimport (richcmp, richcmp_not_equal,
         rich_to_bool, rich_to_bool_sgn)
 
@@ -82,7 +82,7 @@ from sage.rings.real_double import is_RealDoubleField, RDF
 from sage.rings.complex_double import is_ComplexDoubleField, CDF
 from sage.rings.real_mpfi import is_RealIntervalField
 
-from sage.structure.element import generic_power
+from sage.structure.element import generic_power, coerce_binop
 from sage.structure.element cimport (parent, have_same_parent,
         Element, RingElement, coercion_model)
 
@@ -2023,8 +2023,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
         else:
             return self.roots(ring=ring, multiplicities=False)[0]
 
-
-    def __truediv__(self, right):
+    def __truediv__(left, right):
         """
         EXAMPLES::
 
@@ -2089,20 +2088,35 @@ cdef class Polynomial(CommutativeAlgebraElement):
             Traceback (most recent call last):
             ...
             ZeroDivisionError: division by zero in finite field
+
+        Check that :trac:`23611` is fixed::
+
+            sage: int(1) / x
+            1/x
         """
-        try:
-            if not isinstance(right, Element) or right.parent() != self.parent():
-                R = self.parent().base_ring()
+        # Same parents => bypass coercion
+        if have_same_parent(left, right):
+            return (<Element>left)._div_(right)
+
+        # Try division of polynomial by a scalar
+        if isinstance(left, Polynomial):
+            R = (<Polynomial>left)._parent._base
+            try:
                 x = R._coerce_(right)
-                return self * ~x
-        except (TypeError, ValueError):
-            pass
-        return RingElement.__div__(self, right)
+                return left * ~x
+            except TypeError:
+                pass
 
-    def __div__(self, other):
-        return PyNumber_TrueDivide(self, other)
+        # Delegate to coercion model. The line below is basically
+        # RingElement.__truediv__(left, right), except that it also
+        # works if left is not of type RingElement.
+        return wrapperdescr_fastcall(RingElement.__truediv__,
+                left, (right,), <object>NULL)
 
-    def __pow__(self, right, modulus):
+    def __div__(left, right):
+        return PyNumber_TrueDivide(left, right)
+
+    def __pow__(left, right, modulus):
         """
         EXAMPLES::
 
@@ -2127,6 +2141,13 @@ cdef class Polynomial(CommutativeAlgebraElement):
             ...
             TypeError: non-integral exponents not supported
 
+        ::
+
+            sage: int(1)^x
+            Traceback (most recent call last):
+            ...
+            TypeError: unsupported operand type(s) for ** or pow(): 'int' and 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'
+
         ::
 
             sage: x^x
@@ -2183,6 +2204,10 @@ cdef class Polynomial(CommutativeAlgebraElement):
             sage: pow(y*x+1, 51, x^7)
             18009460*y^6*x^6 + 2349060*y^5*x^5 + ... + 51*y*x + 1
         """
+        if not isinstance(left, Polynomial):
+            return NotImplemented
+        cdef Polynomial self = <Polynomial>left
+
         if type(right) is not Integer:
             try:
                 right = Integer(right)
@@ -2200,7 +2225,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
                modulus.leading_coefficient().is_one():
                 return self.power_trunc(right, modulus.degree())
             return power_mod(self, right, modulus)
-        if (<Polynomial>self).is_gen():   # special case x**n should be faster!
+        if self.is_gen():   # special case x**n should be faster!
             P = self.parent()
             R = P.base_ring()
             if P.is_sparse():
diff --git a/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pxd b/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pxd
index e15c5d5f26b..264f05f3c0d 100644
--- a/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pxd
+++ b/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pxd
@@ -23,11 +23,13 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
     cdef zz_pX_c x
     cdef ntl_zz_pContext_class c
     cdef Polynomial_dense_modn_ntl_zz _new(self)
+    cpdef _mod_(self, right)
 
 cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
     cdef ZZ_pX_c x
     cdef ntl_ZZ_pContext_class c
     cdef Polynomial_dense_modn_ntl_ZZ _new(self)
+    cpdef _mod_(self, right)
 
 cdef class Polynomial_dense_mod_p(Polynomial_dense_mod_n):
     pass
diff --git a/src/sage/rings/polynomial/polynomial_quotient_ring.py b/src/sage/rings/polynomial/polynomial_quotient_ring.py
index db0cd092632..8e509dcba98 100644
--- a/src/sage/rings/polynomial/polynomial_quotient_ring.py
+++ b/src/sage/rings/polynomial/polynomial_quotient_ring.py
@@ -343,7 +343,7 @@ def __init__(self, ring, polynomial, name=None, category=None):
 
     def __reduce__(self):
         """
-        TEST:
+        TESTS:
 
         Note the polynomial quotient rings are not unique parent structures::
 
@@ -1362,13 +1362,13 @@ def S_units(self, S, proof=True):
             Unit group with structure C6 of Number Field in a with defining polynomial x^2 + 3
             sage: K.<a> = QQ['x'].quotient(x^2 + 3)
             sage: u,o = K.S_units([])[0]; u, o
-            (1/2*a + 1/2, 6)
+            (-1/2*a + 1/2, 6)
             sage: u^6
             1
             sage: u^3
             -1
             sage: u^2
-            1/2*a - 1/2
+            -1/2*a - 1/2
 
         ::
 
@@ -1377,19 +1377,19 @@ def S_units(self, S, proof=True):
             sage: L.<b> = K['y'].quotient(y^3 + 5); L
             Univariate Quotient Polynomial Ring in b over Number Field in a with defining polynomial x^2 + 3 with modulus y^3 + 5
             sage: L.S_units([])
-            [(1/2*a + 1/2, 6),
+            [(-1/2*a + 1/2, 6),
              ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
              (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
             sage: L.S_units([K.ideal(1/2*a - 3/2)])
             [((-1/6*a - 1/2)*b^2 + (1/3*a - 1)*b + 4/3*a, +Infinity),
-             (1/2*a + 1/2, 6),
+             (-1/2*a + 1/2, 6),
              ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
              (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
             sage: L.S_units([K.ideal(2)])
             [((1/2*a - 1/2)*b^2 + (a + 1)*b + 3, +Infinity),
              ((1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a + 1/2, +Infinity),
              ((1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a - 1/2, +Infinity),
-             (1/2*a + 1/2, 6),
+             (-1/2*a + 1/2, 6),
              ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
              (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
 
@@ -1449,13 +1449,13 @@ def units(self, proof=True):
             Unit group with structure C6 of Number Field in a with defining polynomial x^2 + 3
             sage: K.<a> = QQ['x'].quotient(x^2 + 3)
             sage: u = K.units()[0][0]; u
-            1/2*a + 1/2
+            -1/2*a + 1/2
             sage: u^6
             1
             sage: u^3
             -1
             sage: u^2
-            1/2*a - 1/2
+            -1/2*a - 1/2
             sage: K.<a> = QQ['x'].quotient(x^2 + 5)
             sage: K.units(())
             [(-1, 2)]
@@ -1467,7 +1467,7 @@ def units(self, proof=True):
             sage: L.<b> = K['y'].quotient(y^3 + 5); L
             Univariate Quotient Polynomial Ring in b over Number Field in a with defining polynomial x^2 + 3 with modulus y^3 + 5
             sage: L.units()
-            [(1/2*a + 1/2, 6),
+            [(-1/2*a + 1/2, 6),
              ((-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2, +Infinity),
              (2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2, +Infinity)]
             sage: L.<b> = K.extension(y^3 + 5)
diff --git a/src/sage/rings/polynomial/polynomial_rational_flint.pxd b/src/sage/rings/polynomial/polynomial_rational_flint.pxd
index bfc101d9d06..055d7842668 100644
--- a/src/sage/rings/polynomial/polynomial_rational_flint.pxd
+++ b/src/sage/rings/polynomial/polynomial_rational_flint.pxd
@@ -14,6 +14,7 @@ cdef class Polynomial_rational_flint(Polynomial):
     cdef fmpq_poly_t __poly
 
     cdef Polynomial_rational_flint _new(self)
+    cpdef _mod_(self, right)
     cpdef _unsafe_mutate(self, unsigned long n, value)
     cpdef Polynomial truncate(self, long n)
 
diff --git a/src/sage/rings/polynomial/polynomial_ring_homomorphism.pyx b/src/sage/rings/polynomial/polynomial_ring_homomorphism.pyx
index 45de93b2fb0..18140a24981 100644
--- a/src/sage/rings/polynomial/polynomial_ring_homomorphism.pyx
+++ b/src/sage/rings/polynomial/polynomial_ring_homomorphism.pyx
@@ -94,3 +94,31 @@ cdef class PolynomialRingHomomorphism_from_base(RingHomomorphism_from_base):
             return P({a: f(b) for a, b in x.dict().iteritems()}, *args, **kwds)
         else:
             return P([f(b) for b in x], *args, **kwds)
+
+    def is_injective(self):
+        r"""
+        Return whether this morphism is injective.
+
+        EXAMPLES::
+
+            sage: R.<x> = ZZ[]
+            sage: S.<x> = QQ[]
+            sage: R.hom(S).is_injective()
+            True
+
+        """
+        return self.underlying_map().is_injective()
+
+    def is_surjective(self):
+        r"""
+        Return whether this morphism is surjective.
+
+        EXAMPLES::
+
+            sage: R.<x> = ZZ[]
+            sage: S.<x> = Zmod(2)[]
+            sage: R.hom(S).is_surjective()
+            True
+
+        """
+        return self.underlying_map().is_surjective()
diff --git a/src/sage/rings/polynomial/polynomial_template.pxi b/src/sage/rings/polynomial/polynomial_template.pxi
index adf529cbd6d..3929be1755e 100644
--- a/src/sage/rings/polynomial/polynomial_template.pxi
+++ b/src/sage/rings/polynomial/polynomial_template.pxi
@@ -209,7 +209,7 @@ cdef class Polynomial_template(Polynomial):
             sage: P.<x> = GF(2)[]
             sage: del x
 
-        TEST:
+        TESTS:
 
         The following has been a problem in a preliminary version of
         :trac:`12313`::
diff --git a/src/sage/rings/polynomial/polynomial_template_header.pxi b/src/sage/rings/polynomial/polynomial_template_header.pxi
index 7e369122dcc..64ab106f314 100644
--- a/src/sage/rings/polynomial/polynomial_template_header.pxi
+++ b/src/sage/rings/polynomial/polynomial_template_header.pxi
@@ -7,3 +7,4 @@ from sage.rings.polynomial.polynomial_element cimport Polynomial
 cdef class Polynomial_template(Polynomial):
     cdef celement x
     cdef cparent _cparent
+    cpdef _mod_(self, right)
diff --git a/src/sage/rings/polynomial/real_roots.pyx b/src/sage/rings/polynomial/real_roots.pyx
index f55d7f2d113..0d0b73aafa8 100644
--- a/src/sage/rings/polynomial/real_roots.pyx
+++ b/src/sage/rings/polynomial/real_roots.pyx
@@ -130,7 +130,7 @@ from https://wiki.sagemath.org/days4schedule .
 # This may be vastly faster than the exact calculations carried out
 # by this algorithm!  Is it enough faster to be faster than, say,
 # Pari's floating-point algorithms?)
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from copy import copy
 from random import Random
diff --git a/src/sage/rings/polynomial/skew_polynomial_element.pxd b/src/sage/rings/polynomial/skew_polynomial_element.pxd
index 931122ff43f..6d73ec293e5 100644
--- a/src/sage/rings/polynomial/skew_polynomial_element.pxd
+++ b/src/sage/rings/polynomial/skew_polynomial_element.pxd
@@ -11,6 +11,8 @@ cdef class SkewPolynomial(AlgebraElement):
     cdef SkewPolynomial _new_c(self,list coeffs,Parent P,char check=*)
     cpdef SkewPolynomial _new_constant_poly(self,RingElement a,Parent P,char check=*)
     cpdef _neg_(self)
+    cpdef _floordiv_(self, right)
+    cpdef _mod_(self, right)
 
     cpdef bint is_zero(self)
     cpdef bint is_one(self)
@@ -27,6 +29,8 @@ cdef class SkewPolynomial_generic_dense(SkewPolynomial):
     cdef list _coeffs
 
     cdef void __normalize(self)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
 
     cpdef dict dict(self)
     cpdef list list(self, bint copy=*)
diff --git a/src/sage/rings/polynomial/skew_polynomial_ring.py b/src/sage/rings/polynomial/skew_polynomial_ring.py
index 96c7042f16d..ed9e2904e4e 100644
--- a/src/sage/rings/polynomial/skew_polynomial_ring.py
+++ b/src/sage/rings/polynomial/skew_polynomial_ring.py
@@ -545,7 +545,7 @@ def change_var(self, var):
 
         OUTPUT:
 
-        ``self`` with variable name name changed to ``var``.
+        ``self`` with variable name changed to ``var``.
 
         EXAMPLES::
 
diff --git a/src/sage/rings/polynomial/symmetric_reduction.pyx b/src/sage/rings/polynomial/symmetric_reduction.pyx
index f3e1db1c6cf..3948b551306 100644
--- a/src/sage/rings/polynomial/symmetric_reduction.pyx
+++ b/src/sage/rings/polynomial/symmetric_reduction.pyx
@@ -116,7 +116,7 @@ Symmetric Reduction Strategy is created::
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 import copy
 import operator
diff --git a/src/sage/rings/polynomial/term_order.py b/src/sage/rings/polynomial/term_order.py
index c169f3332f8..9b393ad5242 100644
--- a/src/sage/rings/polynomial/term_order.py
+++ b/src/sage/rings/polynomial/term_order.py
@@ -628,7 +628,7 @@ def __init__(self, name='lex', n=0, force=False):
            constructed. However, it is useful if block orders are
            to be constructed from this ``TermOrder`` object later.
 
-        TEST:
+        TESTS:
 
         We demonstrate that non-positive weights are refused and non-integral weights
         are converted to integers (and potentially rounded)::
@@ -1766,7 +1766,7 @@ def singular_str(self):
             //                  : names    x8 x9
             //        block   4 : ordering C
 
-        TEST:
+        TESTS:
 
         The 'degneglex' ordering is somehow special, it looks like a block
         ordering in SINGULAR.
@@ -1820,7 +1820,7 @@ def singular_moreblocks(self):
             sage: T.singular_moreblocks()
             2
 
-        TEST:
+        TESTS:
 
         The 'degneglex' ordering is somehow special: SINGULAR handles it
         using an extra weight vector block.
diff --git a/src/sage/rings/power_series_ring_element.pxd b/src/sage/rings/power_series_ring_element.pxd
index daa28150b74..067c4f3c6d4 100644
--- a/src/sage/rings/power_series_ring_element.pxd
+++ b/src/sage/rings/power_series_ring_element.pxd
@@ -8,4 +8,4 @@ cdef class PowerSeries(AlgebraElement):
 
     # UNSAFE, only call from an inplace operator
     # may return a new element if not possible to modify inplace
-    cdef _inplace_truncate(self, long prec)
\ No newline at end of file
+    cdef _inplace_truncate(self, long prec)
diff --git a/src/sage/rings/power_series_ring_element.pyx b/src/sage/rings/power_series_ring_element.pyx
index 28382e69c97..d8b17349175 100644
--- a/src/sage/rings/power_series_ring_element.pyx
+++ b/src/sage/rings/power_series_ring_element.pyx
@@ -961,7 +961,7 @@ cdef class PowerSeries(AlgebraElement):
 
     def valuation_zero_part(self):
         r"""
-        Factor self as as `q^n \cdot (a_0 + a_1 q + \cdots)` with
+        Factor self as `q^n \cdot (a_0 + a_1 q + \cdots)` with
         `a_0` nonzero. Then this function returns
         `a_0 + a_1 q + \cdots` .
 
diff --git a/src/sage/rings/rational.pxd b/src/sage/rings/rational.pxd
index 221206f41a0..a04e21f62d0 100644
--- a/src/sage/rings/rational.pxd
+++ b/src/sage/rings/rational.pxd
@@ -8,6 +8,8 @@ cpdef rational_power_parts(a, b, factor_limit=?)
 cdef class Rational(sage.structure.element.FieldElement):
     cdef mpq_t value
 
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cdef __set_value(self, x, unsigned int base)
     cdef void set_from_mpq(Rational self, mpq_t value)
     cdef _lshift(self, long int exp)
diff --git a/src/sage/rings/rational.pyx b/src/sage/rings/rational.pyx
index bad0a6731fc..23360b17760 100644
--- a/src/sage/rings/rational.pyx
+++ b/src/sage/rings/rational.pyx
@@ -558,7 +558,9 @@ cdef class Rational(sage.structure.element.FieldElement):
             if not base:
                 set_from_Rational(self, x.simplest_rational())
             else:
-                xstr = x.str(base)
+                # Truncate in base 10 to match repr(x).
+                # See https://trac.sagemath.org/ticket/21124
+                xstr = x.str(base, truncate=(base == 10))
                 if '.' in xstr:
                     exp = (len(xstr) - (xstr.index('.') +1))
                     p = base**exp
diff --git a/src/sage/rings/real_arb.pxd b/src/sage/rings/real_arb.pxd
index 34d2407cde9..a9a48d534e4 100644
--- a/src/sage/rings/real_arb.pxd
+++ b/src/sage/rings/real_arb.pxd
@@ -13,5 +13,7 @@ cdef int real_part_of_quadratic_element_to_arb(arb_t res, nfeq.NumberFieldElemen
 cdef class RealBall(RingElement):
     cdef arb_t value
     cdef RealBall _new(self)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cpdef RealIntervalFieldElement _real_mpfi_(self, RealIntervalField_class parent)
     cpdef RealBall psi(self)
diff --git a/src/sage/rings/real_arb.pyx b/src/sage/rings/real_arb.pyx
index 6968a26b410..cf89e278c27 100644
--- a/src/sage/rings/real_arb.pyx
+++ b/src/sage/rings/real_arb.pyx
@@ -188,6 +188,7 @@ Classes and Methods
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 from cysignals.signals cimport sig_on, sig_str, sig_off
 
diff --git a/src/sage/rings/real_double.pxd b/src/sage/rings/real_double.pxd
index d76f3bdf6a4..772d4c37d78 100644
--- a/src/sage/rings/real_double.pxd
+++ b/src/sage/rings/real_double.pxd
@@ -11,6 +11,8 @@ cdef class RealDoubleField_class(Field):
 cdef class RealDoubleElement(FieldElement):
     cdef double _value
     cdef _new_c(self, double value)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cpdef RealDoubleElement abs(RealDoubleElement self)
     cdef RealDoubleElement __pow_float(self, double exponent)
     cdef RealDoubleElement __pow_int(self, int exponent)
diff --git a/src/sage/rings/real_double.pyx b/src/sage/rings/real_double.pyx
index b6a5817b914..99bbad36354 100644
--- a/src/sage/rings/real_double.pyx
+++ b/src/sage/rings/real_double.pyx
@@ -40,7 +40,7 @@ Test NumPy conversions::
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 cimport libc.math
 from libc.string cimport memcpy
@@ -339,12 +339,12 @@ cdef class RealDoubleField_class(Field):
         if S is int or S is float:
             return ToRDF(S)
 
-        from rational_field import QQ
-        from real_lazy import RLF
+        from .rational_field import QQ
+        from .real_lazy import RLF
         if S is ZZ or S is QQ or S is RLF:
             return ToRDF(S)
 
-        from real_mpfr import RR, RealField_class
+        from .real_mpfr import RR, RealField_class
         if isinstance(S, RealField_class):
             if S.prec() >= 53:
                 return ToRDF(S)
@@ -424,7 +424,7 @@ cdef class RealDoubleField_class(Field):
         if prec == 53:
             return self
         else:
-            from real_mpfr import RealField
+            from .real_mpfr import RealField
             return RealField(prec)
 
 
diff --git a/src/sage/rings/real_lazy.pxd b/src/sage/rings/real_lazy.pxd
index 0c401aed05d..4110a83ae36 100644
--- a/src/sage/rings/real_lazy.pxd
+++ b/src/sage/rings/real_lazy.pxd
@@ -5,6 +5,8 @@ cdef class LazyField(Field):
     cpdef interval_field(self, prec=*)
 
 cdef class LazyFieldElement(FieldElement):
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cdef LazyFieldElement _new_wrapper(self, value)
     cdef LazyFieldElement _new_binop(self, LazyFieldElement left, LazyFieldElement right, op)
     cdef LazyFieldElement _new_unop(self, LazyFieldElement arg, op)
diff --git a/src/sage/rings/real_lazy.pyx b/src/sage/rings/real_lazy.pyx
index 6c366e87314..df5a331c9dc 100644
--- a/src/sage/rings/real_lazy.pyx
+++ b/src/sage/rings/real_lazy.pyx
@@ -19,7 +19,7 @@ specified in the forward direction).
 #  the License, or (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-
+from __future__ import absolute_import
 
 import math
 
@@ -826,7 +826,7 @@ cdef class LazyFieldElement(FieldElement):
         try:
             return self.eval(complex)
         except Exception:
-            from complex_field import ComplexField
+            from .complex_field import ComplexField
             return complex(self.eval(ComplexField(53)))
 
     cpdef eval(self, R):
@@ -1192,7 +1192,7 @@ cdef class LazyBinop(LazyFieldElement):
         """
         For pickling.
 
-        TEST:
+        TESTS:
             sage: from sage.rings.real_lazy import LazyBinop
             sage: a = LazyBinop(CLF, 3, 2, operator.div)
             sage: loads(dumps(a)) == a
diff --git a/src/sage/rings/real_mpfi.pxd b/src/sage/rings/real_mpfi.pxd
index 51f2c069d31..125d97539d5 100644
--- a/src/sage/rings/real_mpfi.pxd
+++ b/src/sage/rings/real_mpfi.pxd
@@ -40,6 +40,8 @@ cdef class RealIntervalFieldElement(RingElement):
     cdef inline RealIntervalFieldElement _new(self):
         """Return a new real interval with same parent as ``self``."""
         return RealIntervalFieldElement.__new__(RealIntervalFieldElement, self._parent)
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cdef RealIntervalFieldElement abs(RealIntervalFieldElement self)
     cdef Rational _simplest_rational_helper(self)
     cpdef _str_question_style(self, int base, int error_digits, e, bint prefer_sci)
diff --git a/src/sage/rings/real_mpfi.pyx b/src/sage/rings/real_mpfi.pyx
index f782b4147cc..bd370071fb6 100644
--- a/src/sage/rings/real_mpfi.pyx
+++ b/src/sage/rings/real_mpfi.pyx
@@ -1598,7 +1598,7 @@ cdef class RealIntervalFieldElement(RingElement):
 
         Now, consider the precisions needed to represent the endpoints
         (this is the precision that would be produced by
-        ``v.lower().str(no_sci=False, truncate=False)``). Our
+        ``v.lower().str(no_sci=False)``). Our
         result is no more precise than the less precise endpoint, and is
         sufficiently imprecise that the error can be represented with the
         given number of decimal digits. Our result is the most precise
@@ -1685,8 +1685,8 @@ cdef class RealIntervalFieldElement(RingElement):
             style = 'brackets'
 
         if style == 'brackets':
-            t1 = self.lower().str(base=base, no_sci=no_sci, e=e, truncate=False)
-            t2 = self.upper().str(base=base, no_sci=no_sci, e=e, truncate=False)
+            t1 = self.lower().str(base=base, no_sci=no_sci, e=e)
+            t2 = self.upper().str(base=base, no_sci=no_sci, e=e)
 
             return "[%s .. %s]"%(t1, t2)
 
@@ -2211,7 +2211,7 @@ cdef class RealIntervalFieldElement(RingElement):
         EXAMPLES::
 
             sage: R = RealIntervalField(13)
-            sage: R.pi().lower().str(truncate=False)
+            sage: R.pi().lower().str()
             '3.1411'
 
         ::
@@ -2262,7 +2262,7 @@ cdef class RealIntervalFieldElement(RingElement):
         EXAMPLES::
 
             sage: R = RealIntervalField(13)
-            sage: R.pi().upper().str(truncate=False)
+            sage: R.pi().upper().str()
             '3.1417'
 
         ::
diff --git a/src/sage/rings/real_mpfr.pxd b/src/sage/rings/real_mpfr.pxd
index ea319216ba8..a0bdacafed3 100644
--- a/src/sage/rings/real_mpfr.pxd
+++ b/src/sage/rings/real_mpfr.pxd
@@ -21,6 +21,9 @@ cdef class RealNumber(sage.structure.element.RingElement):
     cdef inline RealNumber _new(self):
         """Return a new real number with same parent as ``self``."""
         return <RealNumber>(RealNumber.__new__(RealNumber, self._parent))
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
+    cpdef _mod_(self, right)
     cdef _set(self, x, int base)
     cdef _set_from_GEN_REAL(self, GEN g)
     cdef RealNumber abs(RealNumber self)
diff --git a/src/sage/rings/real_mpfr.pyx b/src/sage/rings/real_mpfr.pyx
index ca12655cbc1..8727a81e757 100644
--- a/src/sage/rings/real_mpfr.pyx
+++ b/src/sage/rings/real_mpfr.pyx
@@ -1389,7 +1389,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
             sage: magma(RealField(1000)(1/3)) # indirect, optional - magma
             0.3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333
         """
-        real_string = self.str(truncate=False)
+        real_string = self.str()
         digit_precision_upper_bound = len(real_string)
         return "%s!%sp%s" % (self.parent()._magma_init_(magma),
                              real_string, digit_precision_upper_bound)
@@ -1540,7 +1540,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
             sage: RR(2.1) # indirect doctest
             2.10000000000000
         """
-        return self.str(10)
+        return self.str(truncate=True)
 
     def _latex_(self):
         r"""
@@ -1553,7 +1553,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
             sage: latex(RR(2e100)) # indirect doctest
             2.00000000000000 \times 10^{100}
         """
-        s = self.str()
+        s = repr(self)
         parts = s.split('e')
         if len(parts) > 1:
             # scientific notation
@@ -1583,7 +1583,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
             sage: s1 == RR(gp(s1))
             True
         """
-        return self.str(10, no_sci=True, truncate=False)
+        return self.str(10, no_sci=True)
 
     def _sage_input_(self, sib, coerced):
         r"""
@@ -1673,8 +1673,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
         # in a number with at least the number of bits we need.
         will_convert = (coerced == 2 or not coerced)
 
-        self_str = self.str(truncate=False,
-                            skip_zeroes=(will_convert or self.prec() <= 53))
+        self_str = self.str(skip_zeroes=(will_convert or self.prec() <= 53))
 
         # To use choice 2 or choice 4, we must be able to read
         # numbers of this precision as a literal.  We support this
@@ -1702,7 +1701,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
                 s = self_str
             else:
                 # This is tricky.  str() uses mpfr_get_str() with
-                # reqdigits=0; this guarantees to give enough digits
+                # digits=0; this guarantees to give enough digits
                 # to recreate the input, if we print and read with
                 # round-to-nearest.  However, we are not going to
                 # read with round-to-nearest, so we might need more digits.
@@ -1715,7 +1714,7 @@ cdef class RealNumber(sage.structure.element.RingElement):
                     fld = RealField(self.prec() + 1, rnd='RNDU')
                 else:
                     fld = RealField(self.prec() + 1, rnd='RNDD')
-                s = fld(self).str(truncate=False)
+                s = fld(self).str()
             v = sib(self.parent())(sib.float_str(repr(s)))
 
         if negative:
@@ -1786,65 +1785,80 @@ cdef class RealNumber(sage.structure.element.RingElement):
         """
         return ZZ(0)
 
-    def str(self, int base=10, no_sci=None, e=None, int truncate=1, bint skip_zeroes=0):
+    def str(self, int base=10, size_t digits=0, *, no_sci=None,
+            e=None, bint truncate=False, bint skip_zeroes=False):
         """
         Return a string representation of ``self``.
 
         INPUT:
 
-        -  ``base`` -- base for output
+        - ``base`` -- (default: 10) base for output
 
-        -  ``no_sci`` -- if 2, never print using scientific notation; if 1
-           or ``True``, print using scientific notation only for very large or
-           very small numbers; if 0 or ``False`` always print with scientific
-           notation; if ``None`` (the default), print how the parent prints.
+        - ``digits`` -- (default: 0) number of digits to display. When
+          ``digits`` is zero, choose this automatically.
 
-        -  ``e`` -- symbol used in scientific notation; defaults to 'e' for
-           base=10, and '@' otherwise
+        - ``no_sci`` -- if 2, never print using scientific notation; if
+          ``True``, use scientific notation only for very large or very
+          small numbers; if ``False`` always print with scientific
+          notation; if ``None`` (the default), print how the parent
+          prints.
 
-        -  ``truncate`` -- if ``True``, round off the last digits in
-           printing to lessen confusing base-2 roundoff issues.
+        - ``e`` -- symbol used in scientific notation; defaults to 'e' for
+          base=10, and '@' otherwise
 
-        -  ``skip_zeroes`` -- if ``True``, skip trailing zeroes in mantissa
+        - ``truncate`` -- (default: ``False``) if ``True``, round off the
+          last digits in base-10 printing to lessen confusing base-2
+          roundoff issues. This flag may not be used in other bases or
+          when ``digits`` is given.
+
+        - ``skip_zeroes`` -- (default: ``False``) if ``True``, skip
+          trailing zeroes in mantissa
 
         EXAMPLES::
 
             sage: a = 61/3.0; a
             20.3333333333333
-            sage: a.str(truncate=False)
+            sage: a.str()
             '20.333333333333332'
+            sage: a.str(truncate=True)
+            '20.3333333333333'
             sage: a.str(2)
             '10100.010101010101010101010101010101010101010101010101'
             sage: a.str(no_sci=False)
-            '2.03333333333333e1'
+            '2.0333333333333332e1'
             sage: a.str(16, no_sci=False)
             '1.4555555555555@1'
+            sage: a.str(digits=5)
+            '20.333'
+            sage: a.str(2, digits=5)
+            '10100.'
+
             sage: b = 2.0^99
             sage: b.str()
-            '6.33825300114115e29'
+            '6.3382530011411470e29'
             sage: b.str(no_sci=False)
-            '6.33825300114115e29'
+            '6.3382530011411470e29'
             sage: b.str(no_sci=True)
-            '6.33825300114115e29'
+            '6.3382530011411470e29'
             sage: c = 2.0^100
             sage: c.str()
-            '1.26765060022823e30'
+            '1.2676506002282294e30'
             sage: c.str(no_sci=False)
-            '1.26765060022823e30'
+            '1.2676506002282294e30'
             sage: c.str(no_sci=True)
-            '1.26765060022823e30'
+            '1.2676506002282294e30'
             sage: c.str(no_sci=2)
-            '1267650600228230000000000000000.'
+            '1267650600228229400000000000000.'
             sage: 0.5^53
             1.11022302462516e-16
             sage: 0.5^54
             5.55111512312578e-17
             sage: (0.01).str()
-            '0.0100000000000000'
+            '0.010000000000000000'
             sage: (0.01).str(skip_zeroes=True)
             '0.01'
             sage: (-10.042).str()
-            '-10.0420000000000'
+            '-10.042000000000000'
             sage: (-10.042).str(skip_zeroes=True)
             '-10.042'
             sage: (389.0).str(skip_zeroes=True)
@@ -1858,10 +1872,48 @@ cdef class RealNumber(sage.structure.element.RingElement):
             1ekg.00000000
             sage: print((65536.0).str(base=62))
             H32.0000000
-            sage: print((65536.0).str(base=63))
+
+        String conversion respects rounding::
+
+            sage: x = -RR.pi()
+            sage: x.str(digits=1)
+            '-3.'
+            sage: y = RealField(53, rnd="RNDD")(x)
+            sage: y.str(digits=1)
+            '-4.'
+            sage: y = RealField(53, rnd="RNDU")(x)
+            sage: y.str(digits=1)
+            '-3.'
+            sage: y = RealField(53, rnd="RNDZ")(x)
+            sage: y.str(digits=1)
+            '-3.'
+            sage: y = RealField(53, rnd="RNDA")(x)
+            sage: y.str(digits=1)
+            '-4.'
+
+        TESTS::
+
+            sage: x = RR.pi()
+            sage: x.str(base=1)
+            Traceback (most recent call last):
+            ...
+            ValueError: base (=1) must be an integer between 2 and 62
+            sage: x.str(base=63)
             Traceback (most recent call last):
             ...
             ValueError: base (=63) must be an integer between 2 and 62
+            sage: x.str(digits=-10)
+            Traceback (most recent call last):
+            ...
+            OverflowError: can't convert negative value to size_t
+            sage: x.str(base=16, truncate=True)
+            Traceback (most recent call last):
+            ...
+            ValueError: truncate is only supported in base 10
+            sage: x.str(digits=10, truncate=True)
+            Traceback (most recent call last):
+            ...
+            ValueError: cannot truncate when digits is given
         """
         if base < 2 or base > 62:
             raise ValueError("base (=%s) must be an integer between 2 and 62" % base)
@@ -1882,23 +1934,19 @@ cdef class RealNumber(sage.structure.element.RingElement):
             else:
                 e = 'e'
 
-        cdef char *s
-        cdef mp_exp_t exponent
-
-        cdef int reqdigits
-
-        reqdigits = 0
-
-        if base == 10 and truncate:
-
-            # This computes reqdigits == floor(log_{10}(2^(b-1))),
+        if truncate:
+            if base != 10:
+                raise ValueError("truncate is only supported in base 10")
+            if digits:
+                raise ValueError("cannot truncate when digits is given")
+            # This computes digits = floor(log_{10}(2^(b-1))),
             # which is the number of *decimal* digits that are
             # "right", given that the last binary bit of the binary
             # number can be off.  That is, if this real is within a
             # relative error of 2^(-b) of an exact decimal with
-            # reqdigits digits, that decimal will be returned.
+            # `digits` digits, that decimal will be returned.
             # This is equivalent to saying that exact decimals with
-            # reqdigits digits differ by at least 2*2^(-b) (relative).
+            # `digits` digits differ by at least 2*2^(-b) (relative).
 
             # (Depending on the precision and the exact number involved,
             # adjacent exact decimals can differ by far more than 2*2^(-b)
@@ -1907,15 +1955,17 @@ cdef class RealNumber(sage.structure.element.RingElement):
             # This avoids the confusion a lot of people have with the last
             # 1-2 binary digits being wrong due to rounding coming from
             # representing numbers in binary.
-
-            reqdigits = <int>(((<RealField_class>self._parent).__prec - 1) * 0.3010299956)
-            if reqdigits <= 1: reqdigits = 2
+            digits = <size_t>(((<RealField_class>self._parent).__prec - 1) * 0.3010299956)
+            if digits < 2:
+                digits = 2
 
         sig_on()
-        s = mpfr_get_str(<char*>0, &exponent, base, reqdigits,
+        cdef char *s
+        cdef mp_exp_t exponent
+        s = mpfr_get_str(NULL, &exponent, base, digits,
                          self.value, (<RealField_class>self._parent).rnd)
         sig_off()
-        if s == <char*> 0:
+        if s is NULL:
             raise RuntimeError("unable to convert an mpfr number to a string")
         t = str(s)
         mpfr_free_str(s)
@@ -1923,18 +1973,13 @@ cdef class RealNumber(sage.structure.element.RingElement):
         if skip_zeroes:
             t = _re_skip_zeroes.match(t).group(1)
 
-        cdef int digits
-        digits = len(t)
-        if t[0] == "-":
-            digits = digits - 1
-
         if no_sci is None:
             no_sci = not (<RealField_class>self._parent).sci_not
 
-        if no_sci is True and ( abs(exponent-1) >=6 ):
+        if no_sci is True and abs(exponent-1) >= 6:
             no_sci = False
 
-        if no_sci is False:
+        if not no_sci:
             if t[0] == "-":
                 return "-%s.%s%s%s" % (t[1:2], t[2:], e, exponent-1)
             return "%s.%s%s%s" % (t[0], t[1:], e, exponent-1)
@@ -2844,18 +2889,18 @@ cdef class RealNumber(sage.structure.element.RingElement):
 
         EXAMPLES::
 
-            sage: (1.0).nexttoward(2).str(truncate=False)
+            sage: (1.0).nexttoward(2).str()
             '1.0000000000000002'
-            sage: (1.0).nexttoward(RR('-infinity')).str(truncate=False)
+            sage: (1.0).nexttoward(RR('-infinity')).str()
             '0.99999999999999989'
             sage: RR(infinity).nexttoward(0)
             2.09857871646739e323228496            # 32-bit
             5.87565378911159e1388255822130839282  # 64-bit
-            sage: RR(pi).str(truncate=False)
+            sage: RR(pi).str()
             '3.1415926535897931'
-            sage: RR(pi).nexttoward(22/7).str(truncate=False)
+            sage: RR(pi).nexttoward(22/7).str()
             '3.1415926535897936'
-            sage: RR(pi).nexttoward(21/7).str(truncate=False)
+            sage: RR(pi).nexttoward(21/7).str()
             '3.1415926535897927'
         """
         cdef RealNumber other_rn
@@ -2885,9 +2930,9 @@ cdef class RealNumber(sage.structure.element.RingElement):
             8.50969131174084e-1388255822130839284  # 64-bit
             sage: RR('+infinity').nextabove()
             +infinity
-            sage: RR(-sqrt(2)).str(truncate=False)
+            sage: RR(-sqrt(2)).str()
             '-1.4142135623730951'
-            sage: RR(-sqrt(2)).nextabove().str(truncate=False)
+            sage: RR(-sqrt(2)).nextabove().str()
             '-1.4142135623730949'
         """
 
@@ -2911,9 +2956,9 @@ cdef class RealNumber(sage.structure.element.RingElement):
             sage: RR('+infinity').nextbelow()
             2.09857871646739e323228496              # 32-bit
             5.87565378911159e1388255822130839282    # 64-bit
-            sage: RR(-sqrt(2)).str(truncate=False)
+            sage: RR(-sqrt(2)).str()
             '-1.4142135623730951'
-            sage: RR(-sqrt(2)).nextbelow().str(truncate=False)
+            sage: RR(-sqrt(2)).nextbelow().str()
             '-1.4142135623730954'
         """
 
diff --git a/src/sage/rings/semirings/tropical_semiring.pyx b/src/sage/rings/semirings/tropical_semiring.pyx
index f03e2786749..3b05330fd91 100644
--- a/src/sage/rings/semirings/tropical_semiring.pyx
+++ b/src/sage/rings/semirings/tropical_semiring.pyx
@@ -19,7 +19,7 @@ AUTHORS:
 #
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-
+from __future__ import absolute_import
 
 from sage.misc.cachefunc import cached_method
 from sage.structure.parent import Parent
diff --git a/src/sage/rings/tests.py b/src/sage/rings/tests.py
index 0f347febbb5..006c24c3281 100644
--- a/src/sage/rings/tests.py
+++ b/src/sage/rings/tests.py
@@ -181,7 +181,7 @@ def rings0():
          (small_finite_field, 'finite field with cardinality at most 2^16'),
          (quadratic_number_field, 'a quadratic number field'),
          (absolute_number_field, 'an absolute number field of degree at most 10')
-         #(relative_number_field, 'a tower of at most 2 extensions each of degree at most 2')  # relative numbers are totally broken broken -- 4782
+         #(relative_number_field, 'a tower of at most 2 extensions each of degree at most 2')  # relative numbers are totally broken -- 4782
          ]
 
     return v
diff --git a/src/sage/schemes/elliptic_curves/cm.py b/src/sage/schemes/elliptic_curves/cm.py
index f0f7db22c3f..cd6d59dd645 100644
--- a/src/sage/schemes/elliptic_curves/cm.py
+++ b/src/sage/schemes/elliptic_curves/cm.py
@@ -520,7 +520,7 @@ def lb(f):
             # So we consider f=1,2,..., until the first f with lb(f)*h_D > c*h_max.
             # (Note that lb(f) is <= 0 for f=1,2, so nothing special is needed there.)
             #
-            # TODO: Maybe we could do better using a bound for for phi_D(f).
+            # TODO: Maybe we could do better using a bound for phi_D(f).
             #
             f = Integer(1)
             chmax=hmax
diff --git a/src/sage/schemes/elliptic_curves/ell_rational_field.py b/src/sage/schemes/elliptic_curves/ell_rational_field.py
index c09153dc4f6..a545876aafd 100644
--- a/src/sage/schemes/elliptic_curves/ell_rational_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_rational_field.py
@@ -1598,7 +1598,7 @@ def analytic_rank_upper_bound(self,
 
           - ``True`` -- the computation is first run with small and then
             successively larger `\Delta` values up to max_Delta. If at any
-            point the computed bound is 0 (or 1 when when root_number is -1
+            point the computed bound is 0 (or 1 when root_number is -1
             or True), the computation halts and that value is returned;
             otherwise the minimum of the computed bounds is returned.
           - ``False`` -- the computation is run a single time with `\Delta`
diff --git a/src/sage/schemes/elliptic_curves/gal_reps.py b/src/sage/schemes/elliptic_curves/gal_reps.py
index ea3e968eb5a..95f1a96b965 100644
--- a/src/sage/schemes/elliptic_curves/gal_reps.py
+++ b/src/sage/schemes/elliptic_curves/gal_reps.py
@@ -1086,7 +1086,12 @@ def image_type(self, p):
             d = K.absolute_degree()
 
             misc.verbose("field of degree %s.  try to compute Galois group"%(d),2)
+            # If the degree is too big, we have no chance at the Galois
+            # group.  K.galois_group calls is_galois which used to rely on
+            # pari's Galois group computations, so degree < 12
             try:
+                if d > 15:
+                    raise Exception()
                 G = K.galois_group()
             except Exception:
                 self.__image_type[p] = "The image is a group of order %s."%d
diff --git a/src/sage/schemes/elliptic_curves/isogeny_class.py b/src/sage/schemes/elliptic_curves/isogeny_class.py
index 7474e852c69..9b81a70718c 100644
--- a/src/sage/schemes/elliptic_curves/isogeny_class.py
+++ b/src/sage/schemes/elliptic_curves/isogeny_class.py
@@ -990,7 +990,7 @@ def __init__(self, E, algorithm="sage", label=None, empty=False):
             sage: E.isogeny_class(order='database')
             Traceback (most recent call last):
             ...
-            RuntimeError: unable to to find Elliptic Curve defined by y^2 = x^3 + 1001 over Rational Field in the database
+            RuntimeError: unable to find Elliptic Curve defined by y^2 = x^3 + 1001 over Rational Field in the database
             sage: TestSuite(isocls).run()
         """
         self._algorithm = algorithm
@@ -1044,11 +1044,11 @@ def _compute(self):
             try:
                 label = self.E.cremona_label(space=False)
             except RuntimeError:
-                raise RuntimeError("unable to to find %s in the database"%self.E)
+                raise RuntimeError("unable to find %s in the database" % self.E)
             db = sage.databases.cremona.CremonaDatabase()
             curves = db.isogeny_class(label)
             if len(curves) == 0:
-                raise RuntimeError("unable to to find %s in the database"%self.E)
+                raise RuntimeError("unable to find %s in the database" % self.E)
             # All curves will have the same conductor and isogeny class,
             # and there are most 8 of them, so lexicographic sorting is okay.
             self.curves = tuple(sorted(curves, key = lambda E: E.cremona_label()))
diff --git a/src/sage/schemes/elliptic_curves/padics.py b/src/sage/schemes/elliptic_curves/padics.py
index 5aa7826fd75..40cc1a55d06 100644
--- a/src/sage/schemes/elliptic_curves/padics.py
+++ b/src/sage/schemes/elliptic_curves/padics.py
@@ -1116,7 +1116,7 @@ def padic_sigma(self, p, N=20, E2=None, check=False, check_hypotheses=True):
     A = (-X.a1()/2 - A) * f
 
     # Convert to a power series and remove the -1/x term.
-    # Also we artificially bump up the accuracy from N-2 to to N-1 digits;
+    # Also we artificially bump up the accuracy from N-2 to N-1 digits;
     # the constant term needs to be known to N-1 digits, so we compute
     # it directly
     assert A.valuation() == -1 and A[-1] == 1
diff --git a/src/sage/schemes/elliptic_curves/period_lattice_region.pyx b/src/sage/schemes/elliptic_curves/period_lattice_region.pyx
index 349df926f92..f975b21bf33 100644
--- a/src/sage/schemes/elliptic_curves/period_lattice_region.pyx
+++ b/src/sage/schemes/elliptic_curves/period_lattice_region.pyx
@@ -32,7 +32,7 @@ REFERENCES:
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-from __future__ import division
+from __future__ import division, absolute_import
 
 import numpy as np
 cimport numpy as np
diff --git a/src/sage/schemes/generic/algebraic_scheme.py b/src/sage/schemes/generic/algebraic_scheme.py
index 6d4dfbb4960..b0eef97acee 100644
--- a/src/sage/schemes/generic/algebraic_scheme.py
+++ b/src/sage/schemes/generic/algebraic_scheme.py
@@ -175,7 +175,7 @@ def is_AlgebraicScheme(x):
 
     OUTPUT:
 
-    Boolean. Whether ``x`` is an an algebraic scheme, that is, a
+    Boolean. Whether ``x`` is an algebraic scheme, that is, a
     subscheme of an ambient space over a ring defined by polynomial
     equations.
 
@@ -2463,6 +2463,60 @@ class AlgebraicScheme_subscheme_projective(AlgebraicScheme_subscheme):
           x^2 - y*z
     """
 
+    def point(self, v, check=True):
+        """
+        Create a point on this projective subscheme.
+
+        INPUT:
+
+        - ``v`` -- anything that defines a point
+
+        - ``check`` -- boolean (optional, default: ``True``); whether
+          to check the defining data for consistency
+
+        OUTPUT: A point of the subscheme.
+
+        EXAMPLES::
+
+            sage: P2.<x,y,z> = ProjectiveSpace(QQ, 2)
+            sage: X = P2.subscheme([x-y,y-z])
+            sage: X.point([1,1,1])
+            (1 : 1 : 1)
+
+        ::
+
+            sage: P2.<x,y> = ProjectiveSpace(QQ, 1)
+            sage: X = P2.subscheme([y])
+            sage: X.point(infinity)
+            (1 : 0)
+
+        ::
+
+            sage: P.<x,y> = ProjectiveSpace(QQ, 1)
+            sage: X = P.subscheme(x^2+2*y^2)
+            sage: X.point(infinity)
+            Traceback (most recent call last):
+            ...
+            TypeError: Coordinates [1, 0] do not define a point on Closed subscheme
+            of Projective Space of dimension 1 over Rational Field defined by:
+              x^2 + 2*y^2
+        """
+        from sage.rings.infinity import infinity
+        if v is infinity  or\
+          (isinstance(v, (list,tuple)) and len(v) == 1 and v[0] is infinity):
+            if self.ambient_space().dimension_relative() > 1:
+                raise ValueError("%s not well defined in dimension > 1"%v)
+            v = [1, 0]
+        # todo: update elliptic curve stuff to take point_homset as argument
+        from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
+        if is_EllipticCurve(self):
+            try:
+                return self._point(self.point_homset(), v, check=check)
+            except AttributeError:  # legacy code without point_homset
+                return self._point(self, v, check=check)
+
+        return self.point_homset()(v, check=check)
+
     def _morphism(self, *args, **kwds):
         r"""
         Construct a morphism determined by action on points of ``self``.
diff --git a/src/sage/schemes/generic/homset.py b/src/sage/schemes/generic/homset.py
index abb4642c525..ee07e3df7e3 100644
--- a/src/sage/schemes/generic/homset.py
+++ b/src/sage/schemes/generic/homset.py
@@ -394,7 +394,6 @@ def _element_constructor_(self, x, check=True):
 
         raise TypeError("x must be a ring homomorphism, list or tuple")
 
-
 #*******************************************************************
 # Base class for points
 #*******************************************************************
diff --git a/src/sage/schemes/projective/endPN_automorphism_group.py b/src/sage/schemes/projective/endPN_automorphism_group.py
index 37ecb658dce..8550eb9ec48 100644
--- a/src/sage/schemes/projective/endPN_automorphism_group.py
+++ b/src/sage/schemes/projective/endPN_automorphism_group.py
@@ -7,7 +7,7 @@
   "Computing Conjugating Sets and Automorphism Groups of Rational Functions" by
   Xander Faber, Michelle Manes, and Bianca Viray [FMV]_.
 
-- Joao de Faria, Ben Hutz, Bianca Thompson (11-2013): adaption for inclusion in Sage
+- Joao de Faria, Ben Hutz, Bianca Thompson (11-2013): adaptation for inclusion in Sage
 
 """
 
diff --git a/src/sage/schemes/projective/projective_morphism.py b/src/sage/schemes/projective/projective_morphism.py
index 5913992d84c..02c9f1901c8 100644
--- a/src/sage/schemes/projective/projective_morphism.py
+++ b/src/sage/schemes/projective/projective_morphism.py
@@ -3223,12 +3223,12 @@ def critical_point_portrait(self, check=True, embedding=None):
             sage: f.critical_point_portrait()
             Traceback (most recent call last):
             ...
-            TypeError: map be be post-critically finite
+            TypeError: map must be post-critically finite
         """
         #input checking done in is_postcritically_finite
         if check:
             if not self.is_postcritically_finite():
-                raise TypeError("map be be post-critically finite")
+                raise TypeError("map must be post-critically finite")
         if embedding is None:
             F = self.change_ring(QQbar)
         else:
diff --git a/src/sage/schemes/projective/projective_point.py b/src/sage/schemes/projective/projective_point.py
index 1f9f5adbe38..7573c117daa 100644
--- a/src/sage/schemes/projective/projective_point.py
+++ b/src/sage/schemes/projective/projective_point.py
@@ -36,7 +36,6 @@
 from sage.categories.integral_domains import IntegralDomains
 from sage.categories.number_fields import NumberFields
 _NumberFields = NumberFields()
-from sage.rings.infinity import infinity
 from sage.rings.integer_ring import ZZ
 from sage.rings.fraction_field import FractionField
 from sage.rings.morphism import RingHomomorphism_im_gens
@@ -133,16 +132,35 @@ def __init__(self, X, v, check=True):
             sage: P = ProjectiveSpace(1, R.quo(t^2+1))
             sage: P([2*t, 1])
             (2*tbar : 1)
+
+        ::
+
+            sage: P = ProjectiveSpace(ZZ,1)
+            sage: P.point(Infinity)
+            (1 : 0)
+            sage: P(infinity)
+            (1 : 0)
+
+        ::
+
+            sage: P = ProjectiveSpace(ZZ,2)
+            sage: P(Infinity)
+            Traceback (most recent call last):
+            ...
+            ValueError: +Infinity not well defined in dimension > 1
+            sage: P.point(infinity)
+            Traceback (most recent call last):
+            ...
+            ValueError: +Infinity not well defined in dimension > 1
         """
         SchemeMorphism.__init__(self, X)
         if check:
             from sage.schemes.elliptic_curves.ell_point import EllipticCurvePoint_field
             d = X.codomain().ambient_space().ngens()
+
             if is_SchemeMorphism(v) or isinstance(v, EllipticCurvePoint_field):
                 v = list(v)
-            elif v is infinity:
-                v = [0] * (d)
-                v[1] = 1
+
             if not isinstance(v,(list,tuple)):
                 raise TypeError("argument v (= %s) must be a scheme point, list, or tuple"%str(v))
             if len(v) != d and len(v) != d-1:
@@ -1522,7 +1540,7 @@ def is_preperiodic(self, f, err=0.1, return_period=False):
         # however precision issues can occur so we can only tell *not* preperiodic
         # if the value is larger than the error
         if h <= err:
-            # if the canonical height is less than than the
+            # if the canonical height is less than the
             # error, then we suspect preperiodic so check
             # either we can find the cycle or the height is
             # larger than the difference between the canonical height
@@ -1607,6 +1625,26 @@ def __init__(self, X, v, check=True):
             sage: Q=P([2, 1])
             sage: Q[0].parent()
             Finite Field of size 7
+
+        ::
+
+            sage: P = ProjectiveSpace(QQ,1)
+            sage: P.point(Infinity)
+            (1 : 0)
+            sage: P(infinity)
+            (1 : 0)
+
+        ::
+
+            sage: P = ProjectiveSpace(QQ,2)
+            sage: P(infinity)
+            Traceback (most recent call last):
+            ...
+            ValueError: +Infinity not well defined in dimension > 1
+            sage: P.point(infinity)
+            Traceback (most recent call last):
+            ...
+            ValueError: +Infinity not well defined in dimension > 1
         """
         SchemeMorphism.__init__(self, X)
         if check:
@@ -1614,10 +1652,7 @@ def __init__(self, X, v, check=True):
             d = X.codomain().ambient_space().ngens()
             if is_SchemeMorphism(v) or isinstance(v, EllipticCurvePoint_field):
                 v = list(v)
-            elif v is infinity:
-                v = [0] * (d)
-                v[1] = 1
-            if not isinstance(v,(list,tuple)):
+            if not isinstance(v, (list,tuple)):
                 raise TypeError("argument v (= %s) must be a scheme point, list, or tuple"%str(v))
             if len(v) != d and len(v) != d-1:
                 raise TypeError("v (=%s) must have %s components"%(v, d))
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 54fdb39481e..d8322ea1091 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -709,6 +709,56 @@ def _point_homset(self, *args, **kwds):
         """
         return SchemeHomset_points_projective_ring(*args, **kwds)
 
+    def point(self, v, check=True):
+        """
+        Create a point on this projective space.
+
+        INPUT:
+
+        - ``v`` -- anything that defines a point
+
+        - ``check`` -- boolean (optional, default: ``True``); whether
+          to check the defining data for consistency
+
+        OUTPUT: A point of this projective space.
+
+        EXAMPLES::
+
+            sage: P2 = ProjectiveSpace(QQ, 2)
+            sage: P2.point([4,5])
+            (4 : 5 : 1)
+
+        ::
+
+            sage: P = ProjectiveSpace(QQ, 1)
+            sage: P.point(infinity)
+            (1 : 0)
+
+        ::
+
+            sage: P = ProjectiveSpace(QQ, 2)
+            sage: P.point(infinity)
+            Traceback (most recent call last):
+            ...
+            ValueError: +Infinity not well defined in dimension > 1
+
+        ::
+
+            sage: P = ProjectiveSpace(ZZ, 2)
+            sage: P.point([infinity])
+            Traceback (most recent call last):
+             ...
+            ValueError: [+Infinity] not well defined in dimension > 1
+        """
+        from sage.rings.infinity import infinity
+        if v is infinity  or\
+          (isinstance(v, (list,tuple)) and len(v) == 1 and v[0] is infinity):
+            if self.dimension_relative() > 1:
+                raise ValueError("%s not well defined in dimension > 1"%v)
+            v = [1, 0]
+
+        return self.point_homset()(v, check=check)
+
     def _point(self, *args, **kwds):
         """
         Construct a point.
diff --git a/src/sage/schemes/riemann_surfaces/riemann_surface.py b/src/sage/schemes/riemann_surfaces/riemann_surface.py
index 7b0efa8f2b2..c19238699e9 100644
--- a/src/sage/schemes/riemann_surfaces/riemann_surface.py
+++ b/src/sage/schemes/riemann_surfaces/riemann_surface.py
@@ -707,7 +707,7 @@ def _determine_new_w(self, z0, oldw, epsilon):
                 wi-=delta
             # If we run 100 iterations without a result, terminate.
             else:
-                raise ConvergenceError("Newton interation fails to converge after %s iterations"%(j,))
+                raise ConvergenceError("Newton iteration fails to converge after %s iterations" % j)
         return neww
 
     def _newton_iteration(self, z0, oldw, epsilon):
diff --git a/src/sage/schemes/toric/chow_group.py b/src/sage/schemes/toric/chow_group.py
index 4b724c5e3e8..12e713eabb6 100644
--- a/src/sage/schemes/toric/chow_group.py
+++ b/src/sage/schemes/toric/chow_group.py
@@ -490,7 +490,7 @@ def cohomology_class(self):
         If the toric variety is not simplicial, that is, has worse
         than orbifold singularities, there is no way to associate a
         cohomology class of the correct degree. In this case,
-        :meth:`cohomology_class` rasies a ``ValueError``.
+        :meth:`cohomology_class` raises a ``ValueError``.
 
         EXAMPLES::
 
diff --git a/src/sage/stats/hmm/distributions.pyx b/src/sage/stats/hmm/distributions.pyx
index 1870b06503c..9901d305b5c 100644
--- a/src/sage/stats/hmm/distributions.pyx
+++ b/src/sage/stats/hmm/distributions.pyx
@@ -17,6 +17,8 @@ AUTHOR:
 #  The full text of the GPL is available at:
 #                  http://www.gnu.org/licenses/
 #############################################################################
+from __future__ import absolute_import
+
 from cpython.object cimport PyObject_RichCompare
 
 cdef extern from "math.h":
diff --git a/src/sage/structure/category_object.pyx b/src/sage/structure/category_object.pyx
index cdf4bff1c4a..45223ba7bd6 100644
--- a/src/sage/structure/category_object.pyx
+++ b/src/sage/structure/category_object.pyx
@@ -483,7 +483,7 @@ cdef class CategoryObject(SageObject):
         """
         This is used by the variable names context manager.
 
-        TEST:
+        TESTS:
 
         In an old version, it was impossible to temporarily change
         the names if no names were previously assigned. But if one
diff --git a/src/sage/structure/coerce.pyx b/src/sage/structure/coerce.pyx
index e723abc5ffc..a87c92bbab2 100644
--- a/src/sage/structure/coerce.pyx
+++ b/src/sage/structure/coerce.pyx
@@ -73,7 +73,7 @@ see the documentation for :class:`Parent`.
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from cpython.object cimport (PyObject, PyTypeObject,
         PyObject_CallObject, PyObject_RichCompare, Py_TYPE)
diff --git a/src/sage/structure/coerce_actions.pyx b/src/sage/structure/coerce_actions.pyx
index 0e3929a1020..013b2b9f492 100644
--- a/src/sage/structure/coerce_actions.pyx
+++ b/src/sage/structure/coerce_actions.pyx
@@ -11,6 +11,7 @@ Coerce actions
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 import operator
 
diff --git a/src/sage/structure/coerce_dict.pyx b/src/sage/structure/coerce_dict.pyx
index 9f04471c69d..bbd1ca7ab96 100644
--- a/src/sage/structure/coerce_dict.pyx
+++ b/src/sage/structure/coerce_dict.pyx
@@ -41,7 +41,7 @@ used with weak references on the values.
 #  the License, or (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 from cpython.object cimport *
 from cpython.list cimport PyList_New
diff --git a/src/sage/structure/coerce_maps.pyx b/src/sage/structure/coerce_maps.pyx
index deee5e617a3..88a8f0eb5f7 100644
--- a/src/sage/structure/coerce_maps.pyx
+++ b/src/sage/structure/coerce_maps.pyx
@@ -1,12 +1,12 @@
 """
 Coerce maps
 """
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 import re
 import types
 
-from parent import Set_PythonType
+from .parent import Set_PythonType
 from sage.structure.parent cimport Parent
 from sage.structure.element cimport Element
 
@@ -42,7 +42,7 @@ cdef class DefaultConvertMap(Map):
             sage: f.parent()
             Set of Morphisms from Finite Field of size 7 to Finite Field of size 11 in Category of sets with partial maps
 
-        Test that `trac`:23211 is resolved::
+        Test that :trac:`23211` is resolved::
 
             sage: f._is_coercion
             False
diff --git a/src/sage/structure/dynamic_class.py b/src/sage/structure/dynamic_class.py
index 2269bd78e39..3d2ed2fae49 100644
--- a/src/sage/structure/dynamic_class.py
+++ b/src/sage/structure/dynamic_class.py
@@ -123,6 +123,7 @@ class MyPermutation(UniqueRepresentation, PermutationCycleType, PosetElement, Gr
 from sage.misc.classcall_metaclass import ClasscallMetaclass
 from sage.misc.inherit_comparison import InheritComparisonMetaclass, InheritComparisonClasscallMetaclass
 
+
 def dynamic_class(name, bases, cls=None, reduction=None, doccls=None,
                   prepend_cls_bases=True, cache=True):
     r"""
@@ -309,6 +310,10 @@ def dynamic_class(name, bases, cls=None, reduction=None, doccls=None,
     """
     bases = tuple(bases)
     #assert(len(bases) > 0 )
+    try:
+        name = str(name)
+    except UnicodeEncodeError:
+        pass
     assert(isinstance(name, str))
     #    assert(cls is None or issubtype(type(cls), type) or type(cls) is classobj)
     if cache is True:
diff --git a/src/sage/structure/element.pyx b/src/sage/structure/element.pyx
index 63024be546d..757513620d7 100644
--- a/src/sage/structure/element.pyx
+++ b/src/sage/structure/element.pyx
@@ -290,9 +290,6 @@ cdef dict _coerce_op_symbols = dict(
         add='+', sub='-', mul='*', div='/', truediv='/', floordiv='//', mod='%',
         iadd='+', isub='-', imul='*', idiv='/', itruediv='/', ifloordiv='//')
 
-cdef MethodType
-from types import MethodType
-
 from sage.structure.richcmp cimport rich_to_bool
 from sage.structure.coerce cimport py_scalar_to_element
 from sage.structure.parent cimport Parent
diff --git a/src/sage/structure/list_clone.pyx b/src/sage/structure/list_clone.pyx
index cff6398da22..6102fcd9a61 100644
--- a/src/sage/structure/list_clone.pyx
+++ b/src/sage/structure/list_clone.pyx
@@ -1130,7 +1130,7 @@ cdef class ClonableList(ClonableArray):
 
     cpdef remove(self, el):
         """
-        Remove the first occurence of ``el`` from ``self``
+        Remove the first occurrence of ``el`` from ``self``
 
         INPUT: ``el`` - any object
 
diff --git a/src/sage/structure/mutability.pyx b/src/sage/structure/mutability.pyx
index 7702c775cb8..86b2d2eab87 100644
--- a/src/sage/structure/mutability.pyx
+++ b/src/sage/structure/mutability.pyx
@@ -11,7 +11,7 @@ Mutability Cython Implementation
 #  Distributed under the terms of the GNU General Public License (GPL)
 #                  http://www.gnu.org/licenses/
 ##########################################################################
-from __future__ import print_function
+from __future__ import print_function, absolute_import
 
 
 cdef class Mutability:
diff --git a/src/sage/structure/parent.pyx b/src/sage/structure/parent.pyx
index 70eb9f85dd2..45ae214793a 100644
--- a/src/sage/structure/parent.pyx
+++ b/src/sage/structure/parent.pyx
@@ -1813,7 +1813,7 @@ cdef class Parent(sage.structure.category_object.CategoryObject):
 
         TESTS:
 
-        We check that `trac`:23184 has been resolved::
+        We check that :trac:`23184` has been resolved::
 
             sage: QQ['x', 'y']._generic_coerce_map(QQ).category_for()
             Category of unique factorization domains
diff --git a/src/sage/symbolic/expression.pxd b/src/sage/symbolic/expression.pxd
index e49210241eb..1c38335bbbc 100644
--- a/src/sage/symbolic/expression.pxd
+++ b/src/sage/symbolic/expression.pxd
@@ -3,6 +3,8 @@ from sage.structure.element cimport CommutativeRingElement
 
 cdef class Expression(CommutativeRingElement):
     cdef GEx _gobj
+    cpdef _add_(self, other)
+    cpdef _mul_(self, other)
     cdef Expression coerce_in(self, z)
     cpdef object _eval_self(self, R)
     cpdef object _convert(self, R)
diff --git a/src/sage/symbolic/function.pyx b/src/sage/symbolic/function.pyx
index 4f863f67f1f..9ceca3576cb 100644
--- a/src/sage/symbolic/function.pyx
+++ b/src/sage/symbolic/function.pyx
@@ -13,14 +13,14 @@ Classes for symbolic functions
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
-from __future__ import division
+from __future__ import division, absolute_import
 
 from sage.libs.pynac.pynac cimport *
 from sage.rings.integer cimport smallInteger
 from sage.structure.sage_object cimport SageObject
 from sage.structure.element cimport Element, parent
 from .expression cimport new_Expression_from_GEx, Expression
-from ring import SR
+from .ring import SR
 
 from sage.structure.coerce cimport py_scalar_to_element, is_numpy_type, is_mpmath_type
 from sage.structure.element cimport coercion_model
diff --git a/src/sage/symbolic/ring.pyx b/src/sage/symbolic/ring.pyx
index a11236836ec..5fa23dc1273 100644
--- a/src/sage/symbolic/ring.pyx
+++ b/src/sage/symbolic/ring.pyx
@@ -12,6 +12,7 @@ The symbolic ring
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
+from __future__ import absolute_import
 
 from sage.libs.pynac.pynac cimport *
 
@@ -190,7 +191,7 @@ cdef class SymbolicRing(CommutativeRing):
 
             from sage.interfaces.maxima import Maxima
 
-            from subring import GenericSymbolicSubring
+            from .subring import GenericSymbolicSubring
 
             if ComplexField(mpfr_prec_min()).has_coerce_map_from(R):
                 # Almost anything with a coercion into any precision of CC
@@ -1063,7 +1064,7 @@ cdef class SymbolicRing(CommutativeRing):
         """
         if self is not SR:
             raise NotImplementedError('Cannot create subring of %s.' % (self,))
-        from subring import SymbolicSubring
+        from .subring import SymbolicSubring
         return SymbolicSubring(*args, **kwds)
 
 SR = SymbolicRing()
diff --git a/src/sage/tensor/modules/free_module_automorphism.py b/src/sage/tensor/modules/free_module_automorphism.py
index 6f54cbef4fc..d52b861843c 100644
--- a/src/sage/tensor/modules/free_module_automorphism.py
+++ b/src/sage/tensor/modules/free_module_automorphism.py
@@ -1090,7 +1090,7 @@ def matrix(self, basis1=None, basis2=None):
 
         OUTPUT:
 
-        - the matrix representing representing the automorphism ``self`` w.r.t
+        - the matrix representing the automorphism ``self`` w.r.t
           to bases ``basis1`` and ``basis2``; more precisely, the columns of
           this matrix are formed by the components w.r.t. ``basis2`` of
           the images of the elements of ``basis1``.
diff --git a/src/sage/tensor/modules/free_module_morphism.py b/src/sage/tensor/modules/free_module_morphism.py
index ff61bb82644..a8d0a68702d 100644
--- a/src/sage/tensor/modules/free_module_morphism.py
+++ b/src/sage/tensor/modules/free_module_morphism.py
@@ -1066,7 +1066,7 @@ def matrix(self, basis1=None, basis2=None):
 
         OUTPUT:
 
-        - the matrix representing representing the homomorphism ``self`` w.r.t
+        - the matrix representing the homomorphism ``self`` w.r.t
           to bases ``basis1`` and ``basis2``; more precisely, the columns of
           this matrix are formed by the components w.r.t. ``basis2`` of
           the images of the elements of ``basis1``.
diff --git a/src/sage/typeset/character_art_factory.py b/src/sage/typeset/character_art_factory.py
index 9a1c3d53fa2..e70288f0382 100644
--- a/src/sage/typeset/character_art_factory.py
+++ b/src/sage/typeset/character_art_factory.py
@@ -16,7 +16,7 @@
 #
 #                  http://www.gnu.org/licenses/
 #*******************************************************************************
-from six import iteritems, string_types
+from six import iteritems, string_types, text_type, binary_type
 
 from sage.structure.sage_object import SageObject
 
@@ -187,10 +187,10 @@ def build_from_string(self, obj):
             bb
             ccc
         """
-        if self.string_type is unicode and not isinstance(obj, unicode):
-            obj = str(obj).decode('utf-8')
-        if self.string_type is str and not isinstance(obj, str):
-            obj = unicode(obj).encode('utf-8')
+        if self.string_type is text_type and not isinstance(obj, text_type):
+            obj = binary_type(obj).decode('utf-8')
+        if self.string_type is binary_type and not isinstance(obj, binary_type):
+            obj = text_type(obj).encode('utf-8')
         return self.art_type(obj.splitlines())
 
     def build_container(self, content, left_border, right_border):
diff --git a/src/sage/typeset/unicode_art.py b/src/sage/typeset/unicode_art.py
index 6ae6a125a8d..8b463db7db9 100644
--- a/src/sage/typeset/unicode_art.py
+++ b/src/sage/typeset/unicode_art.py
@@ -25,6 +25,7 @@
 from sage.typeset.character_art import CharacterArt
 from sage.typeset.character_art_factory import CharacterArtFactory
 import sage.typeset.symbols as symbol
+from six import text_type
 
 
 class UnicodeArt(CharacterArt):
@@ -49,11 +50,11 @@ class UnicodeArt(CharacterArt):
          π⋅x
         ℯ
     """
-    _string_type = unicode
+    _string_type = text_type
 
 
 _unicode_art_factory = CharacterArtFactory(
-    UnicodeArt, unicode, '_unicode_art_',
+    UnicodeArt, text_type, '_unicode_art_',
     (symbol.unicode_left_parenthesis, symbol.unicode_right_parenthesis),
     (symbol.unicode_left_square_bracket, symbol.unicode_right_square_bracket),
     (symbol.unicode_left_curly_brace, symbol.unicode_right_curly_brace),
diff --git a/src/sage/version.py b/src/sage/version.py
index acc73fbf47f..6ea14e226a0 100644
--- a/src/sage/version.py
+++ b/src/sage/version.py
@@ -1,4 +1,4 @@
 # Sage version information for Python scripts
 # This file is auto-generated by the sage-update-version script, do not edit!
-version = '8.1.beta1'
-date = '2017-08-04'
+version = '8.1.beta2'
+date = '2017-08-13'
diff --git a/src/sage_setup/docbuild/__init__.py b/src/sage_setup/docbuild/__init__.py
index 43156631ed3..6f82dd8d2a3 100644
--- a/src/sage_setup/docbuild/__init__.py
+++ b/src/sage_setup/docbuild/__init__.py
@@ -14,7 +14,8 @@
 from __future__ import print_function
 from six.moves import range
 
-import logging, optparse, os, shutil, subprocess, sys, re
+import optparse, os, shutil, subprocess, sys, re
+import logging, warnings
 
 import sphinx.cmdline
 import sphinx.util.console
@@ -76,7 +77,7 @@ def build_ref_doc(args):
 def builder_helper(type):
     """
     Returns a function which builds the documentation for
-    output type type.
+    output type ``type``.
     """
     def f(self, *args, **kwds):
         output_dir = self._output_dir(type)
@@ -110,7 +111,7 @@ def f(self, *args, **kwds):
             logger.warning("LaTeX file written to {}".format(output_dir))
         else:
             logger.warning(
-                "Build finished.  The built documents can be found in {}".
+                "Build finished. The built documents can be found in {}".
                 format(output_dir))
 
     f.is_output_format = True
@@ -481,7 +482,8 @@ def __init__(self, name, lang='en'):
 
     def _output_dir(self, type, lang='en'):
         """
-        Returns the directory where the output of type type is stored.
+        Return the directory where the output of type ``type`` is stored.
+
         If the directory does not exist, then it will automatically be
         created.
 
@@ -653,12 +655,12 @@ def get_all_documents(self, refdir):
 
 class ReferenceSubBuilder(DocBuilder):
     """
-    This class builds sub-components of the reference manual.  It is
-    responsible for making sure the auto generated ReST files for the
+    This class builds sub-components of the reference manual. It is
+    responsible for making sure that the auto generated reST files for the
     Sage library are up to date.
 
     When building any output, we must first go through and check
-    to see if we need to update any of the autogenerated ReST
+    to see if we need to update any of the autogenerated reST
     files.  There are two cases where this would happen:
 
     1. A new module gets added to one of the toctrees.
@@ -684,53 +686,60 @@ def _wrapper(self, build_type, *args, **kwds):
         This is the wrapper around the builder_helper methods that
         goes through and makes sure things are up to date.
         """
-        # Delete the auto-generated .rst files, if the inherited
+        # Force regeneration of all modules if the inherited
         # and/or underscored members options have changed.
         global options
-        inherit_prev = self.get_cache().get('option_inherited')
-        underscore_prev = self.get_cache().get('option_underscore')
-        if (inherit_prev is None or inherit_prev != options.inherited or
-            underscore_prev is None or underscore_prev != options.underscore):
-            logger.info("Detected change(s) in inherited and/or underscored members option(s).")
-            self.clean_auto()
-            self.get_cache.clear_cache()
+        cache = self.get_cache()
+        force = False
+        try:
+            if (cache['option_inherited'] != options.inherited or
+                cache['option_underscore'] != options.underscore):
+                logger.info("Detected change(s) in inherited and/or underscored members option(s).")
+                force = True
+        except KeyError:
+            force = True
+        cache['option_inherited'] = options.inherited
+        cache['option_underscore'] = options.underscore
+        self.save_cache()
 
-        # After "sage -clone", refresh the .rst file mtimes in
+        # After "sage -clone", refresh the reST file mtimes in
         # environment.pickle.
         if options.update_mtimes:
-            logger.info("Checking for .rst file mtimes to update...")
+            logger.info("Checking for reST file mtimes to update...")
             self.update_mtimes()
 
-        #Update the .rst files for modified Python modules
-        logger.info("Updating .rst files with modified modules...")
-        for module_name in self.get_modified_modules():
-            self.write_auto_rest_file(module_name.replace(os.path.sep, '.'))
-
-        #Write the .rst files for newly included modules
-        logger.info("Writing .rst files for newly-included modules...")
-        for module_name in self.get_newly_included_modules(save=True):
-            self.write_auto_rest_file(module_name)
+        if force:
+            # Write reST files for all modules from scratch.
+            self.clean_auto()
+            for module_name in self.get_all_included_modules():
+                self.write_auto_rest_file(module_name)
+        else:
+            # Write reST files for new and updated modules.
+            for module_name in self.get_new_and_updated_modules():
+                self.write_auto_rest_file(module_name)
 
-        #Copy over the custom .rst files from _sage
+        # Copy over the custom reST files from _sage
         _sage = os.path.join(self.dir, '_sage')
         if os.path.exists(_sage):
-            logger.info("Copying over custom .rst files from %s ...", _sage)
+            logger.info("Copying over custom reST files from %s ...", _sage)
             shutil.copytree(_sage, os.path.join(self.dir, 'sage'))
 
         getattr(DocBuilder, build_type)(self, *args, **kwds)
 
     def cache_filename(self):
         """
-        Returns the filename where the pickle of the dictionary of
-        already generated ReST files is stored.
+        Return the filename where the pickle of the reference cache
+        is stored.
         """
         return os.path.join(self._doctrees_dir(), 'reference.pickle')
 
     @cached_method
     def get_cache(self):
         """
-        Retrieve the cache of already generated ReST files.  If it
-        doesn't exist, then we just return an empty dictionary.  If it
+        Retrieve the reference cache which contains the options previously used
+        by the reference builder.
+
+        If it doesn't exist, then we just return an empty dictionary.  If it
         is corrupted, return an empty dictionary.
         """
         filename = self.cache_filename()
@@ -744,26 +753,21 @@ def get_cache(self):
             logger.debug("Cache file '%s' is corrupted; ignoring it..."% filename)
             cache = {}
         else:
-            logger.debug("Loaded .rst file cache: %s", filename)
+            logger.debug("Loaded the reference cache: %s", filename)
         finally:
             file.close()
         return cache
 
     def save_cache(self):
         """
-        Save the cache of already generated ReST files.
+        Pickle the current reference cache for later retrieval.
         """
         cache = self.get_cache()
-
-        global options
-        cache['option_inherited'] = options.inherited
-        cache['option_underscore'] = options.underscore
-
         from six.moves import cPickle
         file = open(self.cache_filename(), 'wb')
         cPickle.dump(cache, file)
         file.close()
-        logger.debug("Saved .rst file cache: %s", self.cache_filename())
+        logger.debug("Saved the reference cache: %s", self.cache_filename())
 
     def get_sphinx_environment(self):
         """
@@ -786,7 +790,7 @@ class Foo(object):
 
     def update_mtimes(self):
         """
-        Updates the modification times for ReST files in the Sphinx
+        Updates the modification times for reST files in the Sphinx
         environment for this project.
         """
         env = self.get_sphinx_environment()
@@ -794,7 +798,7 @@ def update_mtimes(self):
             import time
             for doc in env.all_docs:
                 env.all_docs[doc] = time.time()
-            logger.info("Updated %d .rst file mtimes", len(env.all_docs))
+            logger.info("Updated %d reST file mtimes", len(env.all_docs))
             # This is the only place we need to save (as opposed to
             # load) Sphinx's pickle, so we do it right here.
             env_pickle = os.path.join(self._doctrees_dir(),
@@ -841,7 +845,6 @@ def get_modified_modules(self):
             logger.info("Sphinx found %d modified modules", len(changed))
         except OSError as err:
             logger.debug("Sphinx failed to determine modified modules: %s", err)
-            self.clean_auto()
             return
         for name in changed:
             # Only pay attention to files in a directory sage/... In
@@ -881,34 +884,74 @@ def get_all_included_modules(self):
             for module in self.get_modules(filename):
                 yield module
 
-    def get_newly_included_modules(self, save=False):
+    def get_new_and_updated_modules(self):
         """
-        Returns an iterator for all modules that appear in the
-        toctrees that don't appear in the cache.
+        Return an iterator for all new and updated modules that appear in
+        the toctrees, and remove obsolete old modules.
         """
-        cache = self.get_cache()
-        new_modules = 0
-        for module in self.get_all_included_modules():
-            if module not in cache:
-                cache[module] = True
-                new_modules += 1
-                yield module
-        logger.info("Found %d newly included modules", new_modules)
-        if save:
-            self.save_cache()
+        env = self.get_sphinx_environment()
+        if env is None:
+            all_docs = {}
+        else:
+            all_docs = env.all_docs
+
+        new_modules = []
+        updated_modules = []
+        old_modules = []
+        for module_name in self.get_all_included_modules():
+            docname = module_name.replace('.', os.path.sep)
+
+            if docname not in all_docs:
+                new_modules.append(module_name)
+                yield module_name
+                continue
 
-    def print_newly_included_modules(self):
+            # get the modification timestamp of the reST doc for the module
+            mtime = all_docs[docname]
+            try:
+                with warnings.catch_warnings():
+                    # primarily intended to ignore deprecation warnings
+                    warnings.simplefilter("ignore")
+                    __import__(module_name)
+            except ImportError as err:
+                logger.error("Warning: Could not import %s %s", module_name, err)
+                raise
+            newtime = os.path.getmtime(sys.modules[module_name].__file__)
+
+            if newtime > mtime:
+                updated_modules.append(module_name)
+                yield module_name
+            else: # keep good old module
+                old_modules.append(module_name)
+
+        removed_modules = []
+        for docname in all_docs.keys():
+            if docname.startswith('sage' + os.path.sep):
+                module_name = docname.replace(os.path.sep, '.')
+                if not (module_name in old_modules or module_name in updated_modules):
+                    try:
+                        os.remove(os.path.join(self.dir, docname) + '.rst')
+                    except OSError: # already removed
+                        pass
+                    logger.debug("Deleted auto-generated reST file %s".format(docname))
+                    removed_modules.append(module_name)
+
+        logger.info("Found %d new modules", len(new_modules))
+        logger.info("Found %d updated modules", len(updated_modules))
+        logger.info("Removed %d obsolete modules", len(removed_modules))
+
+    def print_new_and_updated_modules(self):
         """
-        Prints all of the modules that appear in the toctrees that
-        don't appear in the cache.
+        Print all the modules that appear in the toctrees that
+        are newly included or updated.
         """
-        for module_name in self.get_newly_included_modules():
+        for module_name in self.get_new_and_updated_modules():
             print(module_name)
 
     def get_modules(self, filename):
         """
-        Given a filename for a ReST file, return an iterator for
-        all of the autogenerated ReST files that it includes.
+        Given a filename for a reST file, return an iterator for
+        all of the autogenerated reST files that it includes.
         """
         #Create the regular expression used to detect an autogenerated file
         auto_re = re.compile('^\s*(..\/)*(sage(nb)?\/[\w\/]*)\s*$')
@@ -961,7 +1004,7 @@ def auto_rest_filename(self, module_name):
 
     def write_auto_rest_file(self, module_name):
         """
-        Writes the autogenerated ReST file for module_name.
+        Writes the autogenerated reST file for module_name.
         """
         if not module_name.startswith('sage'):
             return
@@ -1001,17 +1044,12 @@ def write_auto_rest_file(self, module_name):
 
     def clean_auto(self):
         """
-        Remove the cache file for the autogenerated files as well as
-        the files themselves.
+        Remove all autogenerated reST files.
         """
-        if os.path.exists(self.cache_filename()):
-            os.unlink(self.cache_filename())
-            logger.debug("Deleted .rst cache file: %s", self.cache_filename())
-
         import shutil
         try:
             shutil.rmtree(os.path.join(self.dir, 'sage'))
-            logger.debug("Deleted auto-generated .rst files in: %s",
+            logger.debug("Deleted auto-generated reST files in: %s",
                          os.path.join(self.dir, 'sage'))
         except OSError:
             pass
@@ -1063,7 +1101,6 @@ def print_included_modules(self):
         for module_name in self.get_all_included_modules():
             print(module_name)
 
-
 class SingleFileBuilder(DocBuilder):
     """
     This is the class used to build the documentation for a single
@@ -1164,7 +1201,8 @@ def setup(app):
 
     def _output_dir(self, type):
         """
-        Returns the directory where the output of type type is stored.
+        Return the directory where the output of type ``type`` is stored.
+
         If the directory does not exist, then it will automatically be
         created.
         """
@@ -1332,8 +1370,8 @@ def help_commands(name='all', s=u""):
     # To do: Generate the lists dynamically, using class attributes,
     # as with the Builders above.
     command_dict = { 'reference' : [
-        'print_included_modules',   'print_modified_modules       (*)',
-        'print_unincluded_modules', 'print_newly_included_modules (*)',
+        'print_included_modules',   'print_modified_modules        (*)',
+        'print_unincluded_modules', 'print_new_and_updated_modules (*)',
         ] }
     for doc in command_dict:
         if name == 'all' or doc == name:
@@ -1493,7 +1531,7 @@ def setup_parser():
                         help="pass comma-separated OPTS to sphinx-build")
     advanced.add_option("-U", "--update-mtimes", dest="update_mtimes",
                         default=False, action="store_true",
-                        help="before building reference manual, update modification times for auto-generated ReST files")
+                        help="before building reference manual, update modification times for auto-generated reST files")
     advanced.add_option("-k", "--keep-going", dest="keep_going",
                         default=False, action="store_true",
                         help="Do not abort on errors but continue as much as possible after an error")
diff --git a/src/sage_setup/docbuild/ext/multidocs.py b/src/sage_setup/docbuild/ext/multidocs.py
index c66ea807251..9a412ee3b16 100644
--- a/src/sage_setup/docbuild/ext/multidocs.py
+++ b/src/sage_setup/docbuild/ext/multidocs.py
@@ -264,13 +264,14 @@ def init_subdoc(app):
             # Master file with indexes computed by merging indexes:
             # Monkey patch index fetching to silence warning about broken index
             def load_indexer(docnames):
-                app.builder.info(bold('Skipping loading of indexes'), nonl=1)
+                app.builder.info(bold('skipping loading of indexes... '), nonl=1)
             app.builder.load_indexer = load_indexer
 
     else:
         app.info(bold("Compiling a sub-document"))
-        app.connect('env-updated', fetch_citation)
         app.connect('html-page-context', fix_path_html)
+        if not app.config.multidoc_first_pass:
+            app.connect('env-updated', fetch_citation)
 
         # Monkey patch copy_static_files to make a symlink to "../"
         def link_static_files():