|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). Cheers, Bill.