function to construct an Elliptic Curve from a plane curve of genus one (using Magma)

[sagetrac-ljpk]

This is a function to turn plane curves of genus 1 with a known point into objects of type EllipticCurve. It relies on MAGMA (as does EllipticCurve_from_cubic) but will add a little functionality. Here is an example of how it works:

sage: x,y,z=MPolynomialRing(QQ,Integer(3),'xyz').gens() # optional
sage: C=Curve(x^4+x^2*y^2-z^4) 
sage: P=C(1,0,1) 
sage: E=EllipticCurve_from_plane_curve(C,P) 
sage: E 
Elliptic Curve defined by y^2  = x^3 + 4*x over Rational Field

Component: algebraic geometry

Author: Lloyd Kilford, John Cremona

Reviewer: Nick Alexander

Merged: sage-4.1.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/4290

[sagetrac-ljpk]

Attachment: EllipticCurve_from_plane_curve.sage.gz

implementation of EllipticCurve_from_plane_curve

[JohnCremona]

comment:3

The "patch" isn't a patch, it's just a function defined in Sage. Secondly, there is already a function that does almost the same as this in sage/schemes/elliptic_curves/constructor.py .

So what needs to be done (Lloyd!) is to (1) Explain briefly how this function differs from the existing one, and why it is better, and also (2) actually make a patch.

If Lloyd convinces me of (1) then I'll happily do (2).

[sagetrac-ljpk]

comment:4

The current Sage function only converts cubics into elliptic curves, but MAGMA can do quite a bit more than that. My function can also cope with other genus one curves such as x^4+x2*y^2-z4 which is not cubic. Possibly the current function should be tweaked slightly so that it can do both cubic and non-cubic examples.

[sagetrac-mabshoff]

comment:5

Anything going on with this patch? It has been sitting around for 4 months :(.

Cheers,

Michael