Bill Allombert on Thu, 06 Aug 2015 15:09:30 +0200

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Re: new GP function ellisomat

On Thu, Aug 06, 2015 at 09:35:16AM +0100, John Cremona wrote:
> Sounds good!  Has this yet been tested on (for example) all the isogeny
> classes in my database?

So far only for conductors <= 9999. I need to optimize the program first.

> Since then I have used a
> different approach -- as Bill knows -- involving no floating point
> arithmetic, and implemented in Sage.

I use the algebraic approach, but I use results of Mazur, Kenku and others
to avoid useless computations (as you note the list of possible isogeny graphs 
is ver short). Specifically, I use Vélu and Kohel formulae for degree 2 and 3,
and Elkies 1998 algorithm for larger prime degrees.
I compute the tree of p-isogenies and combine them if needed.

The Elkies method does not work for a small number of j-invariants where the
differential of the modular equation vanishes. Using Vélu and Kohel formulae
avod this issue.

> Since someone is bound to ask:  in the early years of making tables of
> elliptic curves I was not very systematic about labelling the curves in
> each class, and there is not going to be any simple way of matching the
> curves as computed by Bill's new function to their C. labels, except by
> using ellidentify.

So far, I did not try to order the curve in any particular order, except
that the first curve is always the one given by the user.

> For complete disclosure I have attached \a file I wrote (last touched in
> 2006 it seems) with more details of the historic story. 

Thanks, this is very useful.

Most of my code work also over number fields, except that I do not know
the possible degrees in the exceptionnal case.

Also I am not sure what is the best way to compute the dual isogenies
(from E_i to E).