John Cremona on Mon, 01 Nov 2021 15:48:11 +0100 |
[Date Prev] [Date Next] [Thread Prev] [Thread Next] [Date Index] [Thread Index]
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