Properties

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

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Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $1 - 29 x + 367 x^{2} - 2349 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.117314330344$, $\pm0.262733545783$
Angle rank:  $2$ (numerical)
Number field:  4.0.3244437.1
Galois group:  $D_{4}$
Jacobians:  24

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 24 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4551 42356157 282691850175 1853523431686725 12158003602504751856 79766551113635315013525 523347626757290126395110711 3433683825425614248529597335525 22528399611462981457366098972523575 147808829519910504928186174392518522112

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 53 6455 531935 43058411 3486881378 282429919007 22876792180883 1853020191621971 150094635740211545 12157665467739893630

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.3244437.1.
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.bd_od$2$(not in LMFDB)