Properties

Label 2.81.abi_rj
Base Field $\F_{3^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 17 x + 81 x^{2} )^{2}$
Frobenius angles:  $\pm0.106600758076$, $\pm0.106600758076$
Angle rank:  $1$ (numerical)
Jacobians:  5

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 contains the Jacobians of 5 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4225 41409225 281600035600 1852761401001225 12157750704111390625 79766698031302547865600 523347938799711239069860225 3433684105848095743930383465225 22528399775649284393996212438243600 147808829580277804007459243690942015625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 48 6308 529878 43040708 3486808848 282430439198 22876805821008 1853020342954628 150094636834096758 12157665472705262948

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.ar 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-35}) \)$)$
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.a_aex$2$(not in LMFDB)
2.81.bi_rj$2$(not in LMFDB)
2.81.r_ia$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.a_aex$2$(not in LMFDB)
2.81.bi_rj$2$(not in LMFDB)
2.81.r_ia$3$(not in LMFDB)
2.81.a_ex$4$(not in LMFDB)
2.81.ar_ia$6$(not in LMFDB)