I believe that it's fine in characteristics not 2 or 3. But once you have checked equality of j-invariants (which is of course easy and should be done first) and you have a candidate u, then computing r,s,t appropriately will give a transform which is *guaranteed* to be correct.
My question to you is: suppose that u^12 has more than one 12th root in K. You pick one of them, call it u, and try to construct r,s,t, and you check that the transform then works. But what if one 12th root works and another doesn't? If you try the one which does not work first then you'll miss the isomorphism and give an incorrect negative answer. So it's up to you to prove to me that this does not happen (or change the code). John On 20/12/2007, Robert Bradshaw <[EMAIL PROTECTED]> wrote: > > On Dec 20, 2007, at 3:21 AM, John Cremona wrote: > > > Topic 1: isomorphisms between Weierstrass models > > > > For Robert B as author of weierstrass_morphism.py: It's not a bug but > > I would recommend that you test for equality of j-invariants before > > doing the harder work (which you do by extracting 12th roots of the > > ratio of the discriminants). Similarly in > > > > elliptic_curves/ell_generic.py > > > > in the function is_isomorphic(). > > Good point. > > > Also, I am yet to be convinced that > > the code in weierstrass_morphism.py is correct, but I will reserve > > judgement until I have a counterexample. > > It certainly is *not* correct in char. 2 & 3 since you divide by 2 > > and 3! > > I think I put a note in there about characteristic 2 and 3, but if > not there should be one. Other than that, if I've made a mistake I'll > be happy to be proven wrong, but I'm pretty confident. > > > I have always done this using the ratio of c4 and/or c6 invariants, > > which is fine except in characteristics 2 and 3 anyway, and also > > simpler (since unless j=0 or j=1728 you only need to test for > > something being a square, not need for 12'th roots). That code can > > already be found in simon's ellQ.gp (the new version incorporating > > some improvements I suggested to him). But it does not deal with char > > 2 & 3, which I'll sit down and work through one day, and then submit > > an improved version of weierstrass_morphism.py. > > OK, I'll take a look at that. > > - Robert > > > > -- John Cremona --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
