Properties

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

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $1 - 10 x + 48 x^{2} - 130 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.116678169037$, $\pm0.350288405554$
Angle rank:  $2$ (numerical)
Number field:  4.0.39744.5
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
$A(\F_{q^r})$ 78 27924 4937166 817726416 137654749518 23290419024756 3938061065912142 665500088766649344 112459501937770802334 19005051485687611153524

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 166 2248 28630 370744 4825222 62759428 815833054 10604885524 137859128086

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.39744.5.
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.k_bw$2$2.169.ae_bq