Properties

Label 2.81.abb_mu
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:  $( 1 - 17 x + 81 x^{2} )( 1 - 10 x + 81 x^{2} )$
Frobenius angles:  $\pm0.106600758076$, $\pm0.312505618912$
Angle rank:  $2$ (numerical)
Jacobians:  84

This isogeny class is not 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 84 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4680 42625440 282773855520 1853290193040000 12157627714334037000 79766293202936297717760 523347603971378905253288520 3433683963092832019248575040000 22528399757003972151023316279682080 147808829575634782192479178981954596000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 55 6497 532090 43052993 3486773575 282429005822 22876791184855 1853020265915393 150094636709873050 12157665472323362177

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.ar $\times$ 1.81.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ah_ai$2$(not in LMFDB)
2.81.h_ai$2$(not in LMFDB)
2.81.bb_mu$2$(not in LMFDB)