Properties

Label 2.81.abh_qs
Base Field $\F_{3^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 17 x + 81 x^{2} )( 1 - 16 x + 81 x^{2} )$
Frobenius angles:  $\pm0.106600758076$, $\pm0.151478024726$
Angle rank:  $2$ (numerical)
Jacobians:  0

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 is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4290 41621580 281904634440 1853075318814720 12158005941111137250 79766858524967437824000 523348000830632514498085410 3433684088563591517712280903680 22528399708717801570887258860748360 147808829493376537074427048595644759500

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 49 6341 530452 43048001 3486882049 282431007458 22876808532529 1853020333626881 150094636388168212 12157665465557404901

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.ar $\times$ 1.81.aq 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.ab_aeg$2$(not in LMFDB)
2.81.b_aeg$2$(not in LMFDB)
2.81.bh_qs$2$(not in LMFDB)