[sage-devel] the command: "E = EllipticCurve('14a1') K.<b> = E.division_field(5); K" never finishes

[Steven Craighead]

E = EllipticCurve('14a1')
K.<b> = E.division_field(p); K for p = 2 and 3 finishes quickly. When I
use p = 5 or p = 7, the command never finishes.

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.

[John Cremona]

Post by Steven Craighead
E = EllipticCurve('14a1')
K.<b> = 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.

Post by Steven Craighead
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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