Properties

Label 2.81.abg_pz
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:  $1 - 32 x + 415 x^{2} - 2592 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0549914688267$, $\pm0.208693592337$
Angle rank:  $2$ (numerical)
Number field:  4.0.166032.2
Galois group:  $D_{4}$
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4353 41793153 282055751748 1853075345635977 12157805932625462433 79766501046260613602832 523347582479573769965280897 3433683709351342094908635746313 22528399435189910149350185590589892 147808829345946517727908170670779841473

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 50 6368 530738 43048004 3486824690 282429741734 22876790245394 1853020128981380 150094634565798674 12157665453430897568

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.166032.2.
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.bg_pz$2$(not in LMFDB)