Properties

Label 2.13.aj_bt
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:  $1 - 9 x + 45 x^{2} - 117 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.215685987913$, $\pm0.344616475996$
Angle rank:  $2$ (numerical)
Number field:  4.0.20725.1
Galois group:  $D_{4}$
Jacobians:  3

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 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 89 30349 5131829 828193861 138005338064 23290133646901 3937075508657429 665419951329961509 112455333921596387681 19004890815397948493824

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 179 2333 28995 371690 4825163 62743721 815734819 10604492489 137857962614

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.20725.1.
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.j_bt$2$2.169.j_jx