Properties

Label 2.81.abk_ss
Base Field $\F_{3^{4}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

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

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4096 40960000 280883040256 1851890728960000 12156841915449020416 79765842700025896960000 523347195351541271899672576 3433683501226751295288770560000 22528399312340227300247719345524736 147808829244781290288164388889600000000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 46 6238 528526 43020478 3486548206 282427410718 22876773323086 1853020016664958 150094633747317166 12157665445109791198

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.as 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{4}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{4}}$.

SubfieldPrimitive Model
$\F_{3}$2.3.a_ag
$\F_{3}$2.3.a_g

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.a_agg$2$(not in LMFDB)
2.81.bk_ss$2$(not in LMFDB)
2.81.aj_a$3$(not in LMFDB)
2.81.s_jj$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.a_agg$2$(not in LMFDB)
2.81.bk_ss$2$(not in LMFDB)
2.81.aj_a$3$(not in LMFDB)
2.81.s_jj$3$(not in LMFDB)
2.81.as_gg$4$(not in LMFDB)
2.81.a_gg$4$(not in LMFDB)
2.81.s_gg$4$(not in LMFDB)
2.81.j_dd$5$(not in LMFDB)
2.81.abb_mm$6$(not in LMFDB)
2.81.as_jj$6$(not in LMFDB)
2.81.a_dd$6$(not in LMFDB)
2.81.j_a$6$(not in LMFDB)
2.81.bb_mm$6$(not in LMFDB)
2.81.a_a$8$(not in LMFDB)
2.81.aj_dd$10$(not in LMFDB)
2.81.aj_gg$12$(not in LMFDB)
2.81.a_add$12$(not in LMFDB)
2.81.j_gg$12$(not in LMFDB)