Properties

Label 2.81.abe_op
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:  $1 - 30 x + 379 x^{2} - 2430 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0439844089045$, $\pm0.263626170191$
Angle rank:  $2$ (numerical)
Number field:  4.0.69184.1
Galois group:  $D_{4}$
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4481 42125881 282333399104 1853095978864809 12157570794021386801 79766164027081734071296 523347315301061296337714801 3433683598435900758172770717129 22528399462262267650518070295833664 147808829433613660216210758339712522201

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 52 6420 531262 43048484 3486757252 282428548446 22876778566372 1853020069124804 150094634746167262 12157665460641750900

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.69184.1.
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_op$2$(not in LMFDB)