Properties

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

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Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 9 x )^{2}( 1 - 9 x + 81 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.333333333333$
Angle rank:  $0$ (numerical)
Jacobians:  0

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 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
$A(\F_{q^r})$ 4672 42515200 282428473600 1852737759308800 12157047799607605312 79765842700025896960000 523347304770484649831904832 3433683740526069407785296179200 22528399544939174111650877280774400 147808829371954765040867006688602080000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 55 6481 531442 43040161 3486607255 282427410718 22876778106055 1853020145805121 150094635296999122 12157665455570144401

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.as $\times$ 1.81.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3^{4}}$
The base change of $A$ to $\F_{3^{24}}$ is 1.282429536481.acimic 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 $\F_{3^{24}}$.
Remainder of endomorphism lattice by field

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.aj_a$2$(not in LMFDB)
2.81.j_a$2$(not in LMFDB)
2.81.bb_mm$2$(not in LMFDB)
2.81.a_agg$3$(not in LMFDB)
2.81.a_dd$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.aj_a$2$(not in LMFDB)
2.81.j_a$2$(not in LMFDB)
2.81.bb_mm$2$(not in LMFDB)
2.81.a_agg$3$(not in LMFDB)
2.81.a_dd$3$(not in LMFDB)
2.81.aj_gg$4$(not in LMFDB)
2.81.j_gg$4$(not in LMFDB)
2.81.abk_ss$6$(not in LMFDB)
2.81.as_jj$6$(not in LMFDB)
2.81.aj_a$6$(not in LMFDB)
2.81.s_jj$6$(not in LMFDB)
2.81.bk_ss$6$(not in LMFDB)
2.81.as_gg$12$(not in LMFDB)
2.81.a_add$12$(not in LMFDB)
2.81.a_gg$12$(not in LMFDB)
2.81.s_gg$12$(not in LMFDB)
2.81.a_a$24$(not in LMFDB)
2.81.aj_dd$30$(not in LMFDB)
2.81.j_dd$30$(not in LMFDB)