Properties

Label 2.81.abe_os
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:  $1 - 30 x + 382 x^{2} - 2430 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0930688402366$, $\pm0.249098155797$
Angle rank:  $2$ (numerical)
Number field:  4.0.108400.1
Galois group:  $D_{4}$
Jacobians:  36

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 36 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4484 42167536 282477476324 1853364409286400 12157915990403096804 79766502959280916295536 523347581641754796398026244 3433683771659766410457365606400 22528399562125209699595198299813764 147808829494724457505524837170196424816

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 52 6426 531532 43054718 3486856252 282429748506 22876790208772 1853020162606718 150094635411500452 12157665465668274906

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.108400.1.
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_os$2$(not in LMFDB)