Properties

Label 2.81.abc_nf
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:  $1 - 28 x + 343 x^{2} - 2268 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0378368453396$, $\pm0.309796496498$
Angle rank:  $2$ (numerical)
Number field:  4.0.3632400.1
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4609 42407409 282458416900 1852930711993449 12157261182123833809 79765944927862858592400 523347296814835850353056049 3433683716786073755345243799369 22528399580564708934075595577656900 147808829463530458027810961147215513809

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 54 6464 531498 43044644 3486668454 282427772678 22876777758294 1853020132993604 150094635534352938 12157665463102486304

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.3632400.1.
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_nf$2$(not in LMFDB)