Properties

Label 2.13.ai_bk
Base Field $\F_{13}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $1 - 8 x + 36 x^{2} - 104 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.147614849952$, $\pm0.431019279425$
Angle rank:  $2$ (numerical)
Number field:  4.0.297216.1
Galois group:  $D_{4}$
Jacobians:  4

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 94 29892 4918174 812225424 137766260974 23323623778116 3939358107660622 665463437466931200 112454498783628485566 19004913948662907285252

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 178 2238 28438 371046 4832098 62780094 815788126 10604413734 137858130418

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.297216.1.
All geometric endomorphisms are defined over $\F_{13}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.i_bk$2$2.169.i_abe