Properties

Label 2.13.ai_bi
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:  $1 - 8 x + 34 x^{2} - 104 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.104164352389$, $\pm0.448054596667$
Angle rank:  $2$ (numerical)
Number field:  4.0.1088.2
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 92 29072 4811324 805643264 137559158172 23318043275408 3938936338883708 665444746257907712 112455376305986338268 19005059019349406723472

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 174 2190 28206 370486 4830942 62773374 815765214 10604496486 137859182734

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.1088.2.
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_bi$2$2.169.e_ago