make EllipticCurveHom_composite available from EllipticCurve_field.is…

…ogeny

src/sage/schemes/elliptic_curves/ell_field.py

@@ -767,9 +767,9 @@ def descend_to(self, K, f=None):
             Elist = [E.minimal_model() for E in Elist]
         return Elist
 
-    def isogeny(self, kernel, codomain=None, degree=None, model=None, check=True):
+    def isogeny(self, kernel, codomain=None, degree=None, model=None, check=True, algorithm=None):
         r"""
-        Return an elliptic curve isogeny from self.
+        Return an elliptic-curve isogeny from this elliptic curve.
 
         The isogeny can be determined in two ways, either by a
         polynomial or a set of torsion points.  The methods used are:
@@ -785,6 +785,14 @@ def isogeny(self, kernel, codomain=None, degree=None, model=None, check=True):
           (little endian)) which will define the kernel of the
           isogeny.
 
+        - Factored Isogenies (*experimental* --- see
+          :mod:`sage.schemes.elliptic_curves.hom_composite`):
+          Given a list of points which generate a composite-order
+          subgroup, decomposes the isogeny into prime-degree steps.
+          This can be used to construct isogenies of extremely large,
+          smooth degree.
+          This algorithm is selected using ``algorithm="factored"``.
+
         INPUT:
 
         - ``E``         - an elliptic curve, the domain of the isogeny to
@@ -825,6 +833,11 @@ def isogeny(self, kernel, codomain=None, degree=None, model=None, check=True):
                           kernel polynomial, meaning that its roots
                           are the x-coordinates of a finite subgroup.
 
+        - ``algorithm`` (optional): When ``algorithm="factored"`` is
+          passed, decompose the isogeny into prime-degree steps.
+          The ``degree`` and ``model`` parameters are not supported by
+          ``algorithm="factored"``.
+
         OUTPUT:
 
         An isogeny between elliptic curves. This is a morphism of curves.
@@ -838,11 +851,15 @@ def isogeny(self, kernel, codomain=None, degree=None, model=None, check=True):
             sage: phi.rational_maps()
             ((x^2 + x + 1)/(x + 1), (x^2*y + x)/(x^2 + 1))
 
+        ::
+
             sage: E = EllipticCurve('11a1')
             sage: P = E.torsion_points()[1]
             sage: E.isogeny(P)
             Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field
 
+        ::
+
             sage: E = EllipticCurve(GF(19),[1,1])
             sage: P = E(15,3); Q = E(2,12);
             sage: (P.order(), Q.order())
@@ -852,6 +869,17 @@ def isogeny(self, kernel, codomain=None, degree=None, model=None, check=True):
             sage: phi(E.random_point()) # all points defined over GF(19) are in the kernel
             (0 : 1 : 0)
 
+        ::
+
+            sage: E = EllipticCurve(GF(2^32-5), [170246996, 2036646110])
+            sage: P = E.lift_x(2)
+            sage: E.isogeny(P, algorithm="factored")    # experimental
+            doctest:warning
+            ...
+            Composite morphism of degree 1073721825 = 3^4*5^2*11*19*43*59:
+              From: Elliptic Curve defined by y^2 = x^3 + 170246996*x + 2036646110 over Finite Field of size 4294967291
+              To:   Elliptic Curve defined by y^2 = x^3 + 272790262*x + 1903695400 over Finite Field of size 4294967291
+
         Not all polynomials define a finite subgroup (:trac:`6384`)::
 
             sage: E = EllipticCurve(GF(31),[1,0,0,1,2])
@@ -860,6 +888,8 @@ def isogeny(self, kernel, codomain=None, degree=None, model=None, check=True):
             ...
             ValueError: The polynomial x^3 + 4*x^2 + 27*x + 14 does not define a finite subgroup of Elliptic Curve defined by y^2 + x*y = x^3 + x + 2 over Finite Field of size 31.
 
+        TESTS:
+
         Until the checking of kernel polynomials was implemented in
         :trac:`23222`, the following raised no error but returned an
         invalid morphism.  See also :trac:`11578`::
@@ -873,6 +903,11 @@ def isogeny(self, kernel, codomain=None, degree=None, model=None, check=True):
             ...
             ValueError: The polynomial x^2 + (-396/5*a - 2472/5)*x + 223344/5*a - 196272/5 does not define a finite subgroup of Elliptic Curve defined by y^2 = x^3 + (-13392)*x + (-1080432) over Number Field in a with defining polynomial x^2 - x - 1.
         """
+        if algorithm == "factored":
+            if degree is not None: raise TypeError('algorithm="factored" does not support the "degree" parameter')
+            if model  is not None: raise TypeError('algorithm="factored" does not support the "model" parameter')
+            from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
+            return EllipticCurveHom_composite(self, kernel, codomain=codomain)
         try:
             return EllipticCurveIsogeny(self, kernel, codomain, degree, model, check=check)
         except AttributeError as e: