Re: Transforming general cubic to standard form

[John Cremona <john.cremona@gmail.com>]

These formulas can all be found in Tom Fisher's papers on genus one
models.  The binary quartic case is also in my book.   Note that the
question could mean two different things, given a genus 1 curve C
(e.g. given by one of the types of model you mention):   there is
always an elliptic curve J(C), the Jacobian, whether or not C has any
rational points;   but when C is an n-cover of an elliptic curve E
(with n=3,2,4 respectively in your cases), there is a degree n map
from C to E, and also *if* C has a rational point then C and E are
isomorphic.    To get J(C) you only need the invariants of C (e.g. I
and J of a binary quartic).  The degree n map from C to E, or the
isomorphism from C to E given a rational point on C, are more
complicated.

John Cremona

On Mon, 1 Nov 2021 at 14:19, Grechuk, Bogdan (Dr.) <bg83@leicester.ac.uk> wrote:
>
> Dear Bill Allombert,
>
> I have found online your function
>
> ellfromeqn
>
> that "allows  to  recover  a Weierstrass model for an elliptic curve given by a general plane cubic or by a binary quartic or biquadratic model."
>
> Is it possible to also get the actual transformation of variables?
>
> Sincerely,
> Bogdan