Ticket #6887 (closed enhancement: fixed)
Implement elliptic curve isogenies (continued)
|Reported by:||cremona||Owned by:|
|Component:||elliptic curves||Keywords:||elliptic curve isogeny|
|Cc:||was, wuthrich, shumow, kohel, JCooley||Work issues:|
|Report Upstream:||N/A||Reviewers:||Chris Wuthrich|
|Authors:||John Cremona, Jenny Cooley||Merged in:||sage-4.3.1.alpha0|
Thanks mainly to Dan Shumow, 4.1.1 has some very useful code for constructing elliptic curve isogenies. Together with a summer student Jenny Cooley, I am implementing the following:
- For l=2,3,5,7,13 over any field, find all l-isogenies of a given elliptic curve. (These are the l for which X_0(l) has genus 0).
- For the remaining l for which l-isogenies exist over QQ, similarly.
- Given an elliptic curve over QQ, find the whole isogeny class (this currently exists by wrapping some eclib code, but that it not very robust -- what we are writing will be!)
- Testing if two curves are isogenous (at least over QQ; we can do something over other number fields but I am still working out how to make it rigorous.)
At the moment I am not planning anything over finite fields, where the situation is very different, though the generic code for l=2,3,5,7,13 will work (as it is right now, only as long as the characteristic is not 2, 3 or l, but eventually that will change).
Some of the methods we are implementing were worked out by Mark Watkins and me in an unfinished preprint c.2005.
As one major test of the code for curves over QQ, we are intending to check that the databases are closed under isogeny (as they should be! at least my own should be).
Changed 4 years ago by cremona
- attachment URSS_Poster_Computing_Elliptic_Curve_Isogenies_October_2009.pdf added
comment:18 Changed 3 years ago by mhansen
- Status changed from positive_review to closed
- Resolution set to fixed