ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 27 Sep 2016 07:20:28 -0500Find all n-torsion of an elliptic curvehttp://ask.sagemath.org/question/33055/find-all-n-torsion-of-an-elliptic-curve/I'm sorry if this is a really easy question. I found out how to compute the n-torsion of an elliptic curve $E$ over a given field. But what if I would like to do is find a field $K$ such that all n-torsion points of $E$ are defined over $K$.
In effect compute the n-torsion part of the Tate module. Is this possible with Sagemath? Mon, 11 Apr 2016 03:25:03 -0500http://ask.sagemath.org/question/33055/find-all-n-torsion-of-an-elliptic-curve/Answer by John Cremona for <p>I'm sorry if this is a really easy question. I found out how to compute the n-torsion of an elliptic curve $E$ over a given field. But what if I would like to do is find a field $K$ such that all n-torsion points of $E$ are defined over $K$.</p>
<p>In effect compute the n-torsion part of the Tate module. Is this possible with Sagemath? </p>
http://ask.sagemath.org/question/33055/find-all-n-torsion-of-an-elliptic-curve/?answer=34973#post-id-34973The exact answer to your question is to use the mehod division_field, for example
sage: E = EllipticCurve([0,-1,1,-10,-20])
sage: E.division_field(5,'a')
Number Field in a with defining polynomial x^4 - x^3 + x^2 - x + 1
so that
sage: len(E.change_ring(K).torsion_points())
25
but as Nils said the degree is in general very large -- I chose an example where the degree of the 5-division field is only 4, which is as small as possible over QQ.
Tue, 27 Sep 2016 07:20:28 -0500http://ask.sagemath.org/question/33055/find-all-n-torsion-of-an-elliptic-curve/?answer=34973#post-id-34973Answer by nbruin for <p>I'm sorry if this is a really easy question. I found out how to compute the n-torsion of an elliptic curve $E$ over a given field. But what if I would like to do is find a field $K$ such that all n-torsion points of $E$ are defined over $K$.</p>
<p>In effect compute the n-torsion part of the Tate module. Is this possible with Sagemath? </p>
http://ask.sagemath.org/question/33055/find-all-n-torsion-of-an-elliptic-curve/?answer=33071#post-id-33071this would get you close:
E.torsion_polynomial(n).splitting_field()
You're just some quadratic extensions away from the desired field (you just need to make sure that roots of the division polynomial are indeed x-coordinates of points).
Be careful, though: generally such explicit representations of splitting fields are entirely unworkable because they tend to have very high degrees.Mon, 11 Apr 2016 15:54:35 -0500http://ask.sagemath.org/question/33055/find-all-n-torsion-of-an-elliptic-curve/?answer=33071#post-id-33071