Properties

Label 2.81.abb_mq
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:  $1 - 27 x + 328 x^{2} - 2187 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0728119855854$, $\pm0.323673068911$
Angle rank:  $2$ (numerical)
Number field:  4.0.257725.1
Galois group:  $D_{4}$
Jacobians:  32

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 32 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4676 42570304 282601148816 1853015347540224 12157345287178245476 79766088706538539110400 523347492765511920526165796 3433683905389072304141132104704 22528399711446719288002649309272976 147808829532433331943284638217361822784

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 55 6489 531766 43046609 3486692575 282428281758 22876786323775 1853020234775009 150094636406349526 12157665468769929129

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.257725.1.
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.bb_mq$2$(not in LMFDB)