elliptic curve construction from weierstrass equation

[robertwb]

It would be nice to be able to do

        sage: x, y = var('x,y')
        sage: EllipticCurve(y^2 + y ==  x^3 + x - 9)
        Elliptic Curve defined by y^2 + y = x^3 + x - 9 over Rational Field
        
        sage: R.<x,y> = GF(5)[]
        sage: EllipticCurve(x^3 + x^2 + 2 - y^2 - y*x)
        Elliptic Curve defined by y^2 + x*y  = x^3 + x^2 + 2 over Finite Field of size 5

CC: @rlmill

Component: number theory

Issue created by migration from https://trac.sagemath.org/ticket/5113

[rlmill]

comment:1

Attachment: 5113-ec-construction.patch.gz

I'm curious why you don't just do

a1 = -1*f.coefficient(x*y)
a2 = f.coefficient(x**2)

instead of iterating through f. I'm sure it doesn't matter.

The patch looks good though, positive review.

I've also fixed printing of elliptic curves, see #5118.

[robertwb]

comment:2

I tried that first. The problem is f.coefficient(x) returns everything divisible by one power of x, not the x (alone) term

sage: R.<x,y> = QQ[]
sage: f = x^2 + x*y + y^2*x
sage: f.coef
f.coefficient   f.coefficients  
sage: f.coefficient(x)
y^2 + y
sage: f.coefficient(y^2)
x

It is also harder to exclude bad terms using that method.

[sagetrac-mabshoff]

comment:3

Merged in Sage 3.3.alpha3.

Cheers,

Michael