Properties

Label 2.13.ai_bn
Base Field $\F_{13}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $1 - 8 x + 39 x^{2} - 104 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.207527242884$, $\pm0.398160485086$
Angle rank:  $2$ (numerical)
Number field:  4.0.129168.1
Galois group:  $D_{4}$
Jacobians:  4

This isogeny class is simple and geometrically simple.

Newton polygon

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

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
$A(\F_{q^r})$ 97 31137 5080084 821300649 137854789897 23306063929488 3938303712459073 665433056262766857 112452842730439348468 19004762052047773538097

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 184 2310 28756 371286 4828462 62763294 815750884 10604257566 137857028584

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.129168.1.
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_bn$2$2.169.o_hn