Properties

Label 2.81.abd_og
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:  $( 1 - 16 x + 81 x^{2} )( 1 - 13 x + 81 x^{2} )$
Frobenius angles:  $\pm0.151478024726$, $\pm0.243120792737$
Angle rank:  $2$ (numerical)
Jacobians:  16

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 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4554 42397740 282831106536 1853767579023360 12158284207270731354 79766769864367266912000 523347718290325243344640314 3433683788631360922475376599040 22528399496782612690260060577140456 147808829392230024726910141358664283500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 53 6461 532196 43064081 3486961853 282430693538 22876796182013 1853020171765601 150094634976157796 12157665457237837901

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.aq $\times$ 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 is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.ad_abu$2$(not in LMFDB)
2.81.d_abu$2$(not in LMFDB)
2.81.bd_og$2$(not in LMFDB)