sage-nt Google Group

Women in Sage 4 - Application Deadline May 20

2013-05-01T15:48:39Z

The application for the fourth annual Sage Days for Women is now up:
[link].

This Sage Days will be held July 10 - 15, 2013 in Seattle, Washington.
Participants will stay at the beautiful Shuey
House<[link]>.
The first two days will be a series of introductory level Sage talks on

Re: Elliptic curves over number fields

2013-04-22T15:41:43Z

Review needed at [link]

(and another at [link], also
about elliptic curves over number fields)

John

Elliptic curves over number fields

2013-04-22T09:32:44Z

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

Re: [sage-nt] NT Project/Research Interest Gsoc 2013

2013-04-16T18:04:23Z

Hi David,

Thank you for your reply and concern.

Okie regarding my background in Number Theory, It includes elemantary
subjects like Euclidean algs, modular arithmetic, congruences, euler's
phi, CRT,
factorizations, Fermat theorems, primality checks, Diophantine eqns, and
also Group Theory, RSA, and Encryption Algs. I had a linear algebra course

Re: [sage-nt] NT Project/Research Interest Gsoc 2013

2013-04-15T22:50:29Z

Hi Ahmad,
I'm glad to hear you're interested in working on number theory in Sage.
You mention having taken two number theory classes; what topics did you
cover? I would suggest looking at the following files and folders within
the Sage library and describing how familiar you are with the underlying

Re: [sage-nt] Re: Generators for the Mordell-Weil group of an elliptic curve over Q(sqrt(-5))

2013-04-15T20:05:15Z

You are quite right Nils (of course!) -- I did not mean to suggest
that we got more than a subgroup of finite index. But 2-saturation
would be enough, assuming we start with generators of the full groups
over Q including torsion generators.

Anyone who implements that would have to add the saturation step;

Re: Generators for the Mordell-Weil group of an elliptic curve over Q(sqrt(-5))

2013-04-15T19:21:44Z

A priori that only gets you generators for E(K) tensor_Z Q, though.
The group E(K) is canonically isomorphic to WeilRestriction(E,K/Q)(Q),
which is a (2,2)-isogeny away from E x E^(d). You'd still have to 2-
saturate the group generated by the images in E(K) [which is easy].

Or do we have a theorem somewhere that readily gives us the index of

Re: [sage-nt] Generators for the Mordell-Weil group of an elliptic curve over Q(sqrt(-5))

2013-04-15T13:15:18Z

That makes sense. Thanks!
David

Re: [sage-nt] Generators for the Mordell-Weil group of an elliptic curve over Q(sqrt(-5))

2013-04-15T12:58:32Z

sage: E = EllipticCurve('37a1')
sage: K.<a> = QuadraticField(-5)
sage: F = E.quadratic_twist(-5)
sage: E.rank(), F.rank()
(1, 1)
sage: EK = E.change_ring(K)
sage: gens1 = [EK(P) for P in E.gens()]
sage: iso = F.isomorphism_to(EK)
sage: gens2 = [EK(iso(P)) for P in F.gens()]
sage: gens = gens1+gens2

Re: [sage-nt] Generators for the Mordell-Weil group of an elliptic curve over Q(sqrt(-5))

2013-04-15T12:53:55Z

Here's how. Since E and F are isomorphic over K=Q(sqrt-5) you can
define an isomorphism from F.base_change(K) to E.base_change(K) and
then use that to map points. That gives you "half" the generators
(those in the -1-eigenspace for Galois). For the ones in the
+1-eigenspace just map E.gens() into E.base_change(K).

Re: [sage-nt] Generators for the Mordell-Weil group of an elliptic curve over Q(sqrt(-5))

2013-04-15T11:55:01Z

I guess I don't see how to leverage generators of F over Q into generators
for E(Q(sqrt(-5))). I looked in Silverman's chapter on the Mordell-Weil
group and didn't find anything; a reference would be great.

Of course. Now we just need to get someone to implement it. ;-)
David

NT Project/Research Interest Gsoc 2013

2013-04-14T03:35:34Z

Hello Sage,
I am not sure to whom exactly am I supposed to send this email, but i thought here is a good start.
My name is Ahmad Soliman, I am 3rd year Computer Science student at the German University in Cairo, Egypt. I am willing to join the google summer of code program for 2013.
I have already checked the ideas page for Gsoc, but I have noticed that there was not a separate project related to Number Theory. I loved all the topics available, but I have a certain interest in Number Theory.

Re: [sage-nt] Generators for the Mordell-Weil group of an elliptic curve over Q(sqrt(-5))

2013-04-13T07:51:12Z

That's funny, I thought someone had implemented this: finding E(K)
when K is quadratic and E is defined over Q. It would make a nice
project for someone.

Of course, the first method, using Simon's script, should have worked too.

John

Re: [sage-nt] Generators for the Mordell-Weil group of an elliptic curve over Q(sqrt(-5))

2013-04-12T23:48:28Z

You probably already know this, but you *should* compute with the
quadratic twist. The following will instantly give equivalent
information to what you want:

E = EllipticCurve('522c1')
F = E.quadratic_twist(-5)
F.rank()
F.gens()

Of course, Sage *should* be doing this behind the scenes.

Generators for the Mordell-Weil group of an elliptic curve over Q(sqrt(-5))

2013-04-12T23:36:13Z

I just tried to find E(Q(sqrt(-5))) for a curve of conductor 522 and got a
runtime error. This computation doesn't seem like it would be infeasible;
am I doing something wrong or is my intuition for what's computable out of
whack?
David

sage: E = EllipticCurve('522c1'); E
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 6*x - 54 over Rational