Bill Allombert on Thu, 04 Nov 2021 15:46:35 +0100

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Re: Transforming general cubic to standard form

On Thu, Nov 04, 2021 at 01:54:20PM +0000, Grechuk, Bogdan (Dr.) wrote:
> Dear Bill, John, and everyone,
> Thank you very much for the answers and comments!
> May I have one follow up question? If I have a general genus 1 cubic
> and a rational point exists (so this is elliptic curve), but the
> transformation to Weierstrass form is rational but non-linear, it of
> course does not present the internality of the points. I can use Magma
> or SageMath to find integral points on the Weierstrass model. Is there
> any implementation for finding integer points on the original (general
> cubic) model?

A trick is to inflate the Weierstrass form (by [u,r,s,t]=[D,0,0,0] with
Tate notation) to change the integrality condition by replacing [x,y] by
[x*D^4,y*D^6] where D is the denominator of the mapping.