Properties

Label 2.81.abb_mt
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:  $1 - 27 x + 331 x^{2} - 2187 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0987890015234$, $\pm0.315475092415$
Angle rank:  $2$ (numerical)
Number field:  4.0.19250077.2
Galois group:  $D_{4}$
Jacobians:  24

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 24 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4679 42611653 282730675871 1853221738768533 12157558519443173744 79766245958542830005053 523347583278096840455213759 3433683958361527126079973163077 22528399756037693056694733707909759 147808829574025608338568292471545879808

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 55 6495 532009 43051403 3486753730 282428838543 22876790280301 1853020263362099 150094636703435251 12157665472191003390

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.19250077.2.
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_mt$2$(not in LMFDB)