A related comment, though slightly tangential: for elliptic curves E over QQ we can do E.ap(p) to count points on E mod p without going via the construction of a Sage finite field, which is important for speed. We should have the same for E over number fields:
sage: K.<i> = QuadraticField(-1) sage: E = EllipticCurve([1+i,i,i,0,0]) sage: P = K.prime_above(13) sage: E.reduction(P).trace_of_frobenius() -6 -- works ok, but behind the scenes we have constructed a finite field and a curve over that field which are then discarded. John On 16 September 2014 22:39, Jean-Pierre Flori <[email protected]> wrote: > I must say I'm very excited about the new PARI/GP point counting code. > As a side note I had a quick look at Bills new code and it seems some stuff > we discussed at the last PARI workshop aren't implemented yet. > So you might expect even better stugg if the next six months (another PARI > workshop is planned next January). > I did not look deeply enough to see if Bill uses Mike Harrison tricks to > only deal with integral elements in Kedlaya's algorithm. > > What's sure is that we should also wrap what's already in Sage for Kedlaya > algorithm to be actually used for point counting (the Monsky Washnitzer > stuff has been there for years...) though it should be less efficient than a > C implementation. > And yes the j invariant stuff already present in Sage is crucial as well. > So we should definitely not just call PARI for cardinality of elliptic > curves, but keep some Sage magic around it. > > As far as what to do with the a/b/c options I don't have a clear idea, I'm > always happy to struggle with bleeding edge code, but don't really know how > PARI's API evolves in devel versions and how much of a pain it would be to > ship such a version with Sage. > > -- > You received this message because you are subscribed to the Google Groups > "sage-nt" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > > Visit this group at http://groups.google.com/group/sage-nt. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-nt" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send an email to [email protected]. Visit this group at http://groups.google.com/group/sage-nt. For more options, visit https://groups.google.com/d/optout.
