Properties

Label 2.13.aj_bn
Base field $\F_{13}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $1 - 9 x + 39 x^{2} - 117 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.0228181011636$, $\pm0.419357734967$
Angle rank:  $2$ (numerical)
Number field:  4.0.10933.1
Galois group:  $D_{4}$
Jacobians:  $1$
Isomorphism classes:  1

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $83$ $27805$ $4765943$ $801479125$ $137005100048$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $167$ $2171$ $28059$ $368990$ $4823039$ $62750147$ $815707891$ $10604178533$ $137857322582$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.10933.1.

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_bn$2$2.169.ad_ajn