Properties

Label 2.81.abc_nl
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:  $( 1 - 17 x + 81 x^{2} )( 1 - 11 x + 81 x^{2} )$
Frobenius angles:  $\pm0.106600758076$, $\pm0.290722850198$
Angle rank:  $2$ (numerical)
Jacobians:  56

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 56 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4615 42490305 282727157440 1853383296595545 12157767880071565375 79766362097777811456000 523347568401593656749553135 3433683880049363757802715577705 22528399701917306295297609517831360 147808829578518754283444671557294512625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 54 6476 532002 43055156 3486813774 282429249758 22876789630014 1853020221100196 150094636342860162 12157665472560576476

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.ar $\times$ 1.81.al 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.ag_az$2$(not in LMFDB)
2.81.g_az$2$(not in LMFDB)
2.81.bc_nl$2$(not in LMFDB)