On Sat, 1 Dec 2018 at 08:42, Steven Craighead
wrote:
> E = EllipticCurve('14a1')
> K. = E.division_field(p); K for p = 2 and 3 finishes quickly. When I
> use p = 5 or p = 7, the command never finishes.
>
The docstring warns (do E.division_field? to see it) that "This takes a
very long time when the degree of the
division field is large (e.g. when p is large or when the Galois
representation is surjective)."
Now for this curve
sage: G=E.galois_representation()
sage: G.image_type(5)
'The image is all of GL_2(F_5).'
sage: GL(2,GF(5)).order()
480
so you are constructing a field of degree 480 over Q.
This is done in two steps, first to form the splitting field of the
5-division polynomial :
sage: E.division_polynomial(5)
5*x^12 + 5*x^11 + 280*x^10 - 2095*x^9 - 3765*x^8 - 8550*x^7 - 40320*x^6 +
66490*x^5 - 341085*x^4 + 292705*x^3 - 182920*x^2 + 867445*x - 386051
which will have degree 240, and then there's an additional quadratic
extension on top. So it is not surprising that this takes a lot longer
than the 2- or 3-division fields which are (for this curve) only of degree
2.
>
> I use Sagemath 8.4 on a linux mint 19. This is a toshiba AMD Turion II
> Dual Core Process M500 with 4GB ram. This is a Pentium 64-bit.
>
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