#16942: Construct isogenies graph in elliptic curves
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Reporter: sbesnier | Owner:
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.10
Component: elliptic curves | Resolution:
Keywords: | Merged in:
Authors: Sébastien Besnier | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/ticket/16942 | ff484524a2fb23f947782c46401e8142229cb726
Dependencies: | Stopgaps:
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Comment (by cremona):
Replying to [comment:21 pbruin]:
> Replying to [comment:20 chapoton]:
> > There seems to remain an issue about directed graph versus undirected
graph..
> Yes, and it should also be possible to get the actual elliptic curves in
the isogeny class, not just their ''j''-invariants (see comment:17). It
would probably be a good idea to look at the method
`EllipticCurve_rational_field.isogeny_graph()` and try to keep the
interface consistent with that.
Agreed. There will be differences: over number fields isogeny classes
are finite; in non-CM cases there is a unique degree of cyclic isogeny
between any 2 curves in the class, and the graph we draw uses only those
of prime degree which is enough to connect the graph. In CM cases, the
situation is very similar to ordinary e.c. over finite fields: isogenies
come in 2 types, "horizontal" (between curves whose endomorphism rings are
the same order), and "vertical" (between curves whose endo rings are
different orders in the same imaginary quadratic field). Vertical
isogenies are as before, with a unique cyclic degree; but the horizontal
ones do not have unique degrees, in the sense that if two curves hace CM
by the same imaginary quadratic order, of discriminant D say, then the set
of degrees of isogenies between them is the set of positive integers
represented by some binary quadratic form of discriminant D. I toyed with
the idea of using the quadratic form as a label for the edges in this
case, but in the end went for the smallest prime represented by the form.
Over a finite field, in the ordinary case,the curves in an isogeny class
all have the same number of points and their endo rings are all orders in
the same i.q.field, and one can again distinguish between horizontal and
vertical isogenies. The graph is usually referred to as a "volcano" on
account of its shape, where "vertical" and "horizontal" now have physical
meaning. If coders are up to the challenge we could have some spectacular
isogeny graphs! but getting the layout to look good would be a challenge.
And then there are supersingular curves. Here, for any two isogenous
curves, all but finitely many positive integers arise as isogeny degrees
between them (see the last chapter of Kohel's thesis), and I don't see any
reason for not just showing a complete graph.
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Ticket URL: <http://trac.sagemath.org/ticket/16942#comment:23>
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