Properties

Label 2.81.abe_oq
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:  $1 - 30 x + 380 x^{2} - 2430 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0632554344274$, $\pm0.259213082638$
Angle rank:  $2$ (numerical)
Number field:  4.0.1696576.1
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
$A(\F_{q^r})$ 4482 42139764 282381422562 1853185626758736 12157686905205490002 79766280290773257636564 523347411191392281383748882 3433683668045998059745580160000 22528399511777885815368791796115122 147808829472838339954981605752171670804

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 52 6422 531352 43050566 3486790552 282428960102 22876782757972 1853020106690558 150094635076063252 12157665463868084102

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.1696576.1.
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.be_oq$2$(not in LMFDB)