Re: Transforming general cubic to standard form

Dear John Cremona,

Thank you for the answer.

What I am looking for is the implementation that, given genus 1 equation as an input, produces its Weierstrass model (preferably with integer coefficients) together with actual isomorphism (change of variables). If there are no rational point, the function may return the degree n map you described, but in fact I am ok if it returns just error message that the input has no rational point and is therefore not an elliptic curve. There is a similar function in Magma but the inconvenience is that it requires explicit rational point as an input, so I need first to search for it. I then found function ellfromeqn in PARI that does not require a rational point as an input, and wanted to know if it can output the actual isomorphism (change of variables), but from your answer I understood that not (Because J(C) can be found just from invariants, while isomorphisms are more complicated.)

Sincerely,

Bogdan

From: John Cremona <john.cremona@gmail.com>
Sent: 01 November 2021 14:47
To: Grechuk, Bogdan (Dr.) <bg83@leicester.ac.uk>
Cc: Bill.Allombert@math.u-bordeaux.fr <Bill.Allombert@math.u-bordeaux.fr>; pari-dev@pari.math.u-bordeaux.fr <pari-dev@pari.math.u-bordeaux.fr>
Subject: Re: Transforming general cubic to standard form

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