#6887: Implement elliptic curve isogenies (continued)
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Reporter: cremona | Owner: davidloeffler
Type: enhancement | Status: new
Priority: major | Milestone:
Component: elliptic curves | Keywords: elliptic curve isogeny
Reviewer: | Author:
Merged: |
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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:
1. 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).
2. For the remaining {{{l}}} for which {{{l}}}-isogenies exist over QQ,
similarly.
3. 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!)
4. 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).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6887>
Sage <http://sagemath.org/>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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