Properties

Label 2.81.abj_sa
Base Field $\F_{3^{4}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 9 x )^{2}( 1 - 17 x + 81 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.106600758076$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

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
$A(\F_{q^r})$ 4160 41184000 281241309440 1852326013824000 12157296301288424000 79766270364517761024000 523347567075494241117976640 3433683803537410211490983424000 22528399543994754656095934693423360 147808829412529547052623058652140000000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 47 6273 529202 43030593 3486678527 282428924958 22876789572047 1853020179809793 150094635290706962 12157665458907527073

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.as $\times$ 1.81.ar 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 are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.ab_afo$2$(not in LMFDB)
2.81.b_afo$2$(not in LMFDB)
2.81.bj_sa$2$(not in LMFDB)
2.81.ai_j$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.ab_afo$2$(not in LMFDB)
2.81.b_afo$2$(not in LMFDB)
2.81.bj_sa$2$(not in LMFDB)
2.81.ai_j$3$(not in LMFDB)
2.81.ar_gg$4$(not in LMFDB)
2.81.r_gg$4$(not in LMFDB)
2.81.aba_md$6$(not in LMFDB)
2.81.i_j$6$(not in LMFDB)
2.81.ba_md$6$(not in LMFDB)