On 4/27/07, Prof. J. E. Cremona <[EMAIL PROTECTED]> wrote: > I think you (we) need to work out a sensible order for doing these > things. As far as I know the functionality for elliptic curves over Q > in SAGE comes from my C++ code -- what if anything has been written in > "native" SAGE code?
A lot of functionality comes from the PARI C library, but there's also a lot written in native SAGE code, e.g., the group law, invariants of curves, computation of p-adic regulators and p-adic L-series, and computation of complex L-series. However, there's nothing nontrivial along the lines of Tate's algorithm etc. over number fields yet. > For elliptic curves over number fields, I > implemented Tate's algorithm in Magma and can provide the code for that > too (it is now in the standard Magma release) as I did for the heights. Excellent -- many thanks! > This gives conductors, Tamagawa numbers, local minimal models, ... It > does not use local fields, as instead of using completions at primes it > uses localization instead. In practice all that means is that when > finding a minimal model for a curve at a prime ideal P which may not be > principal, it needs to be provided with a uniformiser pi at P, which is > an element with valuation 1 at P (but which may have nonzero valuation > at other primes if P is not principal). While I remember, the builtin > Magma function for that always gives an integral element pi, which is > actually not the most convenient; for then, pi may have positive > valuations at other primes, and then at the stage in the akgorithm where > the coefficient a_i is divided by pi^i the model becomes nonintegral > (even if it was originally integral). Better would be to take pi to > have negative valuations at other primes if necessary (let I be an ideal > in the same class as P and coprime to P and take pi to be a generator of > the fractional ideal P/I). This is why the output of Magma's > LocalMinimalModel function is nonintegral for nonprincipal primes, in > case anyone is interested! I will fix that before sending the code > (+GPL) to William. Excellent. Thanks. William --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
