Properties

Label 2.81.abd_oa
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:  $1 - 29 x + 364 x^{2} - 2349 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0843104972791$, $\pm0.276447108888$
Angle rank:  $2$ (numerical)
Number field:  4.0.1141272.2
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
$A(\F_{q^r})$ 4548 42314592 282552613632 1853277743837568 12157713899722300548 79766305078966740983808 523347479886540892610608068 3433683777857815302907439973888 22528399625041128513427336385127168 147808829551648849493807957103271412832

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 53 6449 531674 43052705 3486798293 282429047870 22876785760805 1853020165951553 150094635830675450 12157665470350456049

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.1141272.2.
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_oa$2$(not in LMFDB)