Elliptic Curve Enumeration by height

[haikona]

Add functionality to Sage that allows one to enumerate elliptic curves over Q by height for a range of Weierstrass families of curve.

This code has grown from work with Wei Ho, who desired a way to enumerate curves over Q by height (where the height of a curve is a function of the coefficients in its Weierstrass model) so that the average value of some datum (e.g.rank, 2-Selmer size) could be computed for curves up to a given height. This code introduces a class, CurveEnumerator(), that computes lists of curves ordered by height for this purpose.

https://github.com/haikona/CBH

Component: elliptic curves

Keywords: enumeration

Author: Simon Spicer

Branch/Commit: u/chapoton/13436 @ 8721f1d

Issue created by migration from https://trac.sagemath.org/ticket/13436

[haikona]

Attachment: trac_13436_curve_enumerator.patch.gz

[haikona]

comment:1

Version 1.0 patch posted. All new code (except for an added import statement in sage.schemes.elliptic_curves.all) is in a new file, curve_enumerator.py, in the sage.schemes.elliptic_curves directory. It's thoroughly documented and doctested, so hopefully its logic will straightforward enough to follow.

[williamstein]

comment:3

For the three_selmer function, you need a doctest.

sage: magma('2+2')  # optional - magma

Then

sage -t --only_optional=magma curve_enumerator.py 

to test.

[williamstein]

comment:4

Fix some docstring formatitng, e.g., the funny indent:

    def heights(self, lowerbound, upperbound):  
        """
        Return a list of permissable curve heights in the specified range
         (bounds inclusive), and for each height the equation coefficients
         that produce curves of that height.

This should output "100%":

...elliptic_curves$ sage -coverage curve_enumerator.py 
----------------------------------------------------------------------
curve_enumerator.py
ERROR: Please add a `TestSuite(s).run()` doctest.
SCORE curve_enumerator.py: 87% (35 of 40)

Missing documentation:
	 * __init__(self):
	 * __repr__(self):
	 * _coeffs_to_a_invariants(self, c):


Missing doctests:
	 * __init__(self):
	 * three_selmer(self, curves, rank=True, reduced=False, output_filename=None, problems_filename=None, \ proof=True, return_data=True, print_timing=True):

[williamstein]

comment:5

• Here: over Q up I would put `\QQ` or at QQ.

• Either: Or: -- maybe instead a sublist of two things?

• weird formatting:

sage: C = EllipticCurveEnumerator(family="short_weierstrass")
sage: C.n_torsion?

       * "rank"              -- (Default True): Compute the rank versus
         size of the curve's torsion/n-torsion subgroup. If True, the
         n-torsion rank is computed (i.e. 0, 1 or 2). If set to False, the
         size of the n-torsion subgroup is computed instead (i.e. n^rank)
    
         * "output_filename" -- (Default None): If not None, the string
    
         name of the file to which the output will be saved.

• Restructure code so that this is natural:

v = EllipticCurveEnumerator(family="short_weierstrass", height=1000).ranks()

Will need caching:

sage: from sage.misc.cachefunc import cached_method
sage: cached_method?

[ohanar]

comment:6

There are three AssertionErrors that should be ValueErrors.

also two_selmer and three_selmer have an extra space for the OUTPUT portion of the docstring.

[fchapoton]

comment:8

I have uploaded a patch that cleans up the documentation.

Still needs works, I think

[fchapoton]

comment:9

could please someone with magma add the missing doctest for "three_selmer" ?

[JohnCremona]

comment:10

I applied the patches to 5.10 and ran the doctest and report the result here.

sage: C = EllipticCurveEnumerator(family="short_weierstrass") 
sage: L = C.coefficients_over_height_range(4, 4)                                                               
sage: R, P = C.three_selmer(L, rank=True,return_data=True,print_timing=False)                                  
sage: R                                                                                                        
[(4, [0, 0, 0, -1, -2], 1),
 (4, [0, 0, 0, -1, 2], 0),
 (4, [0, 0, 0, 0, -2], 1),
 (4, [0, 0, 0, 0, 2], 1),
 (4, [0, 0, 0, 1, -2], 0),
 (4, [0, 0, 0, 1, 2], 0)]
sage: P                                                                                                        
[]

[fchapoton]

comment:11

Attachment: trac_13436_curve_enumerator-addon1.patch.gz

thanks for the help !

[fchapoton]

Commit: a3af039

[fchapoton]

New commits:

a3af039	trac 13436 : doc corrections
e405730	Add functionality to Sage that allows one to enumerate elliptic curves over Q by height for a range of Weierstrass families of curve.

[fchapoton]

Branch: u/chapoton/13436

[JohnCremona]

comment:14

Thanks for converting this to git and fixing the docstring formatting (which I will trust has been done properly)!

Authors: can you remove the place(s) where there are comments "this should be checked" or "todo"? And in several similar places you can delete the comment that some integers must be Sage integers or ... some rather frivolous consequence will occur?

I see that you have carefully allowed for the unavoidable fact that there will be "problem curves" and have an option output filename for these. But I don't think that any of the doctests illustrate this -- please add one or more doctests which do. Meanwhile I am trying the functions out.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

ba097ec	Merge branch 'u/chapoton/13436' of ssh://trac.sagemath.org:22/sage into 13436
0f75b4f	trac #13436 minor changes after some of the reviewer's comments

[sagetrac-git]

Changed commit from a3af039 to 0f75b4f

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

b4b547b	Merge branch 'u/chapoton/13436' into 6.8.b2
512eb17	trac #13436 fixing one error

[sagetrac-git]

Changed commit from 0f75b4f to 512eb17

[fchapoton]

comment:20

There is still one failing doctest, that I do not understand.

[sagetrac-git]

Changed commit from 512eb17 to 3fc917c

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d4fe9b6	Merge branch 'u/chapoton/13436' into 7.1.b2
73502c0	trac #13436 updating cartesian_products
3fc917c	trac #13436 pep8

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

850d7a3	Merge branch 'u/chapoton/13436' into 7.2.b4
3785184	trac #13426 fixing one import

[sagetrac-git]

Changed commit from 3fc917c to 3785184

[sagetrac-git]

Changed commit from 3785184 to 23fa861

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

23fa861

Merge branch 'u/chapoton/13436' in 7.3.rc0

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

04d7e40	Merge branch 'u/chapoton/13436' in 7.6.b5
8721f1d	trac 13436 details

[sagetrac-git]

Changed commit from 23fa861 to 8721f1d