Properties

Label 2.81.abg_qa
Base Field $\F_{3^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $1 - 32 x + 416 x^{2} - 2592 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0814298088129$, $\pm0.199292490656$
Angle rank:  $2$ (numerical)
Number field:  4.0.147712.3
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4354 41807108 282107019586 1853180164518928 12157958798217988994 79766677498438420393988 523347751625829242736572354 3433683846767748685695415222272 22528399529930320847227930036689154 147808829400343418280885631488839221508

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 50 6370 530834 43050438 3486868530 282430366498 22876797639186 1853020203139454 150094635197003186 12157665457905185890

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.147712.3.
All geometric endomorphisms are defined over $\F_{3^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.bg_qa$2$(not in LMFDB)