Properties

Label 2.81.abc_no
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:  $1 - 28 x + 352 x^{2} - 2268 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.133086535686$, $\pm0.278231506244$
Angle rank:  $2$ (numerical)
Number field:  4.0.5863680.3
Galois group:  $D_{4}$
Jacobians:  48

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 48 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4618 42531780 282861555898 1853607276426000 12158008051894180858 79766532799190357642820 523347630951294933864337738 3433683855509967289081276416000 22528399640994093547434361655434378 147808829523022297969440837011284114500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 54 6482 532254 43060358 3486882654 282429854162 22876792364214 1853020207857278 150094635936961494 12157665467995846802

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.5863680.3.
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_no$2$(not in LMFDB)