Properties

Label 2.81.abc_nn
Base Field $\F_{3^{4}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $1 - 28 x + 351 x^{2} - 2268 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.124262281589$, $\pm0.282730282950$
Angle rank:  $2$ (numerical)
Number field:  4.0.88592.1
Galois group:  $D_{4}$
Jacobians:  4

This isogeny class is simple and geometrically simple.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4617 42517953 282816754308 1853532787754313 12157928970691398057 79766478695865098248848 523347615455109181447945833 3433683871265097726749572691337 22528399669522869732220049926149828 147808829548276655830696851322697565633

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 54 6480 532170 43058628 3486859974 282429662598 22876791686838 1853020216359684 150094636127033418 12157665470073084240

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.88592.1.
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.bc_nn$2$(not in LMFDB)