Properties

Label 2.13.ah_bf
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:  $1 - 7 x + 31 x^{2} - 91 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.171237405357$, $\pm0.464284369344$
Angle rank:  $2$ (numerical)
Number field:  4.0.589541.1
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
$A(\F_{q^r})$ 103 30797 4908259 812332469 138024863248 23335821681341 3938886321816451 665406078106775525 112453276162713512311 19004944842626077700352

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 7 183 2233 28443 371742 4834623 62772577 815717811 10604298439 137858354518

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.589541.1.
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.h_bf$2$2.169.n_z