Properties

Label 2.81.abf_pg
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:  $( 1 - 9 x )^{2}( 1 - 13 x + 81 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.243120792737$
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
$A(\F_{q^r})$ 4416 41952000 282165601536 1853017994112000 12157574551206148416 79766181704571328512000 523347284535421141700840256 3433683503605204513284828672000 22528399332059567325095452566195456 147808829311383034760430255547020000000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 51 6393 530946 43046673 3486758331 282428611038 22876777221531 1853020017948513 150094633878696546 12157665450587960073

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.as $\times$ 1.81.an 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.af_acu$2$(not in LMFDB)
2.81.f_acu$2$(not in LMFDB)
2.81.bf_pg$2$(not in LMFDB)
2.81.ae_bt$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.af_acu$2$(not in LMFDB)
2.81.f_acu$2$(not in LMFDB)
2.81.bf_pg$2$(not in LMFDB)
2.81.ae_bt$3$(not in LMFDB)
2.81.an_gg$4$(not in LMFDB)
2.81.n_gg$4$(not in LMFDB)
2.81.aw_kt$6$(not in LMFDB)
2.81.e_bt$6$(not in LMFDB)
2.81.w_kt$6$(not in LMFDB)