Elliptic curves over number fields

[John Cremona]

Anyone inteerested in elliptic curves over number fields might be
interested to look at trac #14472. I have been working on fixing the
bug reported there which makes certain functions fail for relative
extensions, basically anything which calls the method
_reduced_model(). The issue is that we wish to choose the paramaters
r,s,t in a unimodular change of model (u,r,s,t) with u=1 so that
a1,a2,a3 are "reduced" modulo 2,3,2 in some sense of "reduced".

The old code assumed that (after checking that tha coefficients were
algebraic integers) list(c) for each coefficient c was a list of
integers, which certainly fails for relative extensions, but also for
absolute extensions in general.

(Example: In K=Q(a) with a=sqrt(5) the element c=(1+a)/2 is integral
but list(c)=(1/2,1/2).)

In my new code (not yet posted) I have changed this to use
ZK.coordinates(c) inspead of list(c), where ZK is the ring if
integers. This works fine, but does have the property that it depends
on the integral basis. One bad example, which already showed up in
one doctest (see the ticket) is that if you define K as above then the
integral basis is by default [1/2*a + 1/2, a] , not including 1,
which has the effect that 1 mod 2 is not "reduced", which is not nice.

My current plan is to change that doctest to use the same curve but
define the field uing x^2-x-1. I justify this on the grounds that
anyone working over quadratic fields who cares about how output looks
to this extent will define the field to be generated by an element a
such that 1,a is an integral basis.

Any comments?

I will be posting a patch shortly after more testing.

John

[John Cremona]

Review needed at http://trac.sagemath.org/sage_trac/ticket/14472

(and another at http://trac.sagemath.org/sage_trac/ticket/12509, also
about elliptic curves over number fields)

John