Opened 5 years ago

Last modified 19 months ago

#16942 needs_work enhancement

Construct isogenies graph in elliptic curves

Reported by: sbesnier Owned by:
Priority: major Milestone: sage-6.10
Component: elliptic curves Keywords:
Cc: defeo, cremona, jpflori Merged in:
Authors: Sébastien Besnier Reviewers:
Report Upstream: N/A Work issues:
Branch: public/ticket/16942 (Commits) Commit: 4a943f815ec67eebbf1c63e45a0ed8baf369481c
Dependencies: Stopgaps:

Description (last modified by chapoton)

Let E be an EllipticCurve?.

We can not yet construct the graph of the j-invariants of the curves which are isogenous to E. This ticket fills this little gap.

Change History (26)

comment:1 Changed 5 years ago by sbesnier

  • Branch set to u/sbesnier/ticket/16942
  • Commit set to 777a5c7ca5397ec02a8e6969c8eb49721f843a62

New commits:

0d583b4First try to build the volcanoe of an elliptic curve.
777a5c7First draft for isogenies_graph

comment:2 Changed 5 years ago by git

  • Commit changed from 777a5c7ca5397ec02a8e6969c8eb49721f843a62 to e6f92a663ec25111b411c6da6a6ceb973d0b31e7

Branch pushed to git repo; I updated commit sha1. New commits:

e6f92a6Add some doc tests

comment:3 Changed 5 years ago by chapoton

  • Branch changed from u/sbesnier/ticket/16942 to public/ticket/16942
  • Commit changed from e6f92a663ec25111b411c6da6a6ceb973d0b31e7 to a4740c62600bebd6eed5cc3cc88041404407f14b

I have made a cosmetic cleanup of the branch.

But it does not work, test do not pass for me. The doc should say (and the tests should also care) that this is optional, because requiring a database.


New commits:

167a6baMerge branch 'u/sbesnier/ticket/16942' of ssh://trac.sagemath.org:22/sage into 16942
a4740c6trac #16942 cosmetic cleanup
Last edited 5 years ago by chapoton (previous) (diff)

comment:4 Changed 4 years ago by git

  • Commit changed from a4740c62600bebd6eed5cc3cc88041404407f14b to b4bf82d4129bdfff34dd4d85bfccb683978f9de5

Branch pushed to git repo; I updated commit sha1. New commits:

b4bf82dAdd note about the necessity of having the db_modular_polynomials database.

comment:5 Changed 4 years ago by chapoton

Do you want to put this in the state "needs review" ?

comment:6 Changed 4 years ago by sbesnier

  • Status changed from new to needs_review

I thought I did it... Thank you.

comment:7 Changed 4 years ago by chapoton

  • Status changed from needs_review to needs_info

It seems that db_modular_polynomials is not available using

sage -i db_modular_polynomials

I think it would be good to explain how to install this database in the doc, if there is some other way.

And AttributeError should be ValueError instead.

And .. note:: should be .. NOTE::

Last edited 4 years ago by chapoton (previous) (diff)

comment:8 Changed 4 years ago by git

  • Commit changed from b4bf82d4129bdfff34dd4d85bfccb683978f9de5 to 53fbe4b2ffc9933c0bf8748a9a67a86a3c9671bd

Branch pushed to git repo; I updated commit sha1. New commits:

537097fMerge branch 'public/ticket/16942' into 6.5.b4
53fbe4btrac #16942 details

comment:9 Changed 4 years ago by chapoton

  • Cc cremona added
  • Description modified (diff)
  • Milestone changed from sage-6.4 to sage-6.8

comment:10 Changed 4 years ago by git

  • Commit changed from 53fbe4b2ffc9933c0bf8748a9a67a86a3c9671bd to 5ebe464834daa3e5a0517f6552397893aeffbbb5

Branch pushed to git repo; I updated commit sha1. New commits:

153b354Merge branch 'public/ticket/16942' of trac.sagemath.org:sage into 6.8.rc1
5ebe464trac #16942 no longer depending on database

comment:11 Changed 4 years ago by chapoton

  • Status changed from needs_info to needs_review

This now does no longer depends on any database.

comment:12 Changed 4 years ago by pbruin

Should the method perhaps be renamed isogeny_graph()? I think "isogeny graph" is much more widely used than "isogenies graph" (and Google agrees).

comment:13 Changed 4 years ago by git

  • Commit changed from 5ebe464834daa3e5a0517f6552397893aeffbbb5 to e5b72e8afdaaf726e5191fa5db86ac43994ffe72

Branch pushed to git repo; I updated commit sha1. New commits:

e5b72e8changing name

comment:14 Changed 4 years ago by cremona

I did not understand the purpose of this ticket from its description, but I do now that I read the code. For elliptic curves over Q and other number fields we have a class for isogeny classes, which has (amongst other methods) a method for displaying the graph. So I wonder whether it might be possible to put this new code under the same umbrella. But that would be more work, so I do not want to delay this.

Certainly "isogeny graph" is a better term.

comment:15 Changed 4 years ago by pbruin

I don't have time to really review this now (last working day before my holidays), but why does this return a directed graph? If there is an isogeny in one direction, there always is the dual isogeny in the opposite direction.

comment:16 follow-up: Changed 4 years ago by cremona

Good point. And the behaviour for "l = characteristic" should be documented; probably not allowed?

Using the modular polynomial is one way to do it. You could also make use of the existing method E.isogenies_prime_degree(). But in fact as it stands this function is not a method of elliptic curves at all, the input is really just and element j of a finite field. So surely there could be a separate function (not a class method) with input an element of F_q and a prime l, and output a graph on some subset of Fq, and then the elliptic curve class method would call that on its j-invariant.

comment:17 in reply to: ↑ 16 Changed 4 years ago by pbruin

Replying to cremona:

Using the modular polynomial is one way to do it. You could also make use of the existing method E.isogenies_prime_degree(). But in fact as it stands this function is not a method of elliptic curves at all, the input is really just and element j of a finite field. So surely there could be a separate function (not a class method) with input an element of F_q and a prime l, and output a graph on some subset of Fq, and then the elliptic curve class method would call that on its j-invariant.

Actually, the method should perhaps even (optionally?) return the graph of actual elliptic curves instead of the graph of j-invariants. It is likely that the user will want to know exactly which curves over the base field appear in the isogeny graph; if you just have the j-invariants, you only know them up to twist.

comment:18 Changed 4 years ago by git

  • Commit changed from e5b72e8afdaaf726e5191fa5db86ac43994ffe72 to ff484524a2fb23f947782c46401e8142229cb726

Branch pushed to git repo; I updated commit sha1. New commits:

ff48452trac #16942 check that l is not equal to p (characteristic)

comment:19 Changed 4 years ago by chapoton

  • Milestone changed from sage-6.8 to sage-6.9

comment:20 follow-up: Changed 3 years ago by chapoton

  • Authors changed from sbesnier to Sébastien Besnier
  • Milestone changed from sage-6.9 to sage-6.10

I am not really the man to give a positive review, given that I am not an expert.

There seems to remain an issue about directed graph versus undirected graph..

comment:21 in reply to: ↑ 20 ; follow-up: Changed 3 years ago by pbruin

Replying to 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.

comment:22 Changed 3 years ago by pbruin

  • Status changed from needs_review to needs_work

comment:23 in reply to: ↑ 21 Changed 3 years ago by cremona

Replying to pbruin:

Replying to 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.

comment:24 Changed 2 years ago by jpflori

  • Cc jpflori added

comment:25 Changed 19 months ago by git

  • Commit changed from ff484524a2fb23f947782c46401e8142229cb726 to 4a943f815ec67eebbf1c63e45a0ed8baf369481c

Branch pushed to git repo; I updated commit sha1. New commits:

4a943f8Merge branch 'public/ticket/16942' in 8.1.b3

comment:26 Changed 19 months ago by defeo

Hello. Thanks @chapoton for taking the time to rebase this.

But, honestly, shouldn't we say this code is not going to be merged?

I am sure there is a lot of interest right now in isogeny graphs over finite fields (because post-quantum crypto, you know), and it would be good if Sage offered some functionality. However this code hardly covers all interesting cases, and it has design issues.

If we could agree on the correct API, I'd be happy to rewrite the code. Let me summarize the issues:

  • A function on Fq which outputs a graph labeled by j-invariants, VS a function on elliptic curves which outputs a graph labeled by curves. My opinion is that both are useful, so we could implement both.
  • Undirected VS directed graph. The problem is that, although generically these graphs are simple undirected graphs, there are lots of pathological cases:
    • Self-loops,
    • multi-edges,
    • j-invariants 0 and 1728 are especially nasty (same dual for many isogenies).
  • The method: modular polynomials VS isogenies_prime_degree(). As far as I recall, isogenies_prime_degree() factors the division polynomial. It is more expensive, but it requires no optional database (and can in principle cover any degree, although it becomes unrealistic when degrees get large).

Let me add one more issue: fool-proofness. The size of these graphs is polynomial in the field SIZE! I'm sure a lot of beginners would create a cyptographic-size finite field and then ask for an isogeny graph. This would of course take forever. Maybe we could print a warning when the expected size is too large.

@cremona, @pbruin, comments?

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