Construct isogenies graph in elliptic curves

[Sebastien Besnier]

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.

[Sebastien Besnier]

New commits:

​0d583b4	First try to build the volcanoe of an elliptic curve.
​777a5c7	First draft for isogenies_graph

[git]

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

​e6f92a6

Add some doc tests

[Frédéric Chapoton]

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.

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New commits:

​167a6ba	Merge branch 'u/sbesnier/ticket/16942' of ssh://trac.sagemath.org:22/sage into 16942
​a4740c6	trac #16942 cosmetic cleanup

[git]

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

​b4bf82d

Add note about the necessity of having the db_modular_polynomials database.

[Frédéric Chapoton]

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

[Sebastien Besnier]

I thought I did it... Thank you.

[Frédéric Chapoton]

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::

[git]

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

​537097f	Merge branch 'public/ticket/16942' into 6.5.b4
​53fbe4b	trac #16942 details

[git]

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

​153b354	Merge branch 'public/ticket/16942' of trac.sagemath.org:sage into 6.8.rc1
​5ebe464	trac #16942 no longer depending on database

[Frédéric Chapoton]

This now does no longer depends on any database.

[Peter Bruin]

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

[git]

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

​e5b72e8

changing name

[John 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.

[Peter Bruin]

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.

[John 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.

[Peter Bruin]

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.

[git]

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

​ff48452

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

[Frédéric Chapoton]

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..

[Peter Bruin]

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.

[John 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.

[git]

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

​4a943f8

Merge branch 'public/ticket/16942' in 8.1.b3

[Luca De Feo]

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?

[John Cremona]

See my comments on #30199.

[John Cremona]

By the way, Sage's optional database of David Kohel's modular polynomials is tiny. Andrew Sutherland has a very much larger database.

[Luca De Feo]

Maybe we can close this ticket, now that #30199 is going to get in?

Re the modular polynomial database, yes, I had been toying with repackaging it for a while. I hope to be able to finish that work some day...

Maybe it would be good to create a meta-ticket summarizing all wanted functionality regarding isogeny classes and graphs? I can do that if you like.