Properties

Label 2.81.abc_np
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:  $1 - 28 x + 353 x^{2} - 2268 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.142096013077$, $\pm0.273278171236$
Angle rank:  $2$ (numerical)
Number field:  4.0.280225.1
Galois group:  $D_{4}$
Jacobians:  32

This isogeny class is simple and geometrically 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 32 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4619 42545609 282906359600 1853681593837129 12158086157032608859 79766584109891602745600 523347641129637534134885659 3433683832323803761238477838729 22528399604634419165263146081383600 147808829491803805712342190185133579209

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 54 6484 532338 43062084 3486905054 282430035838 22876792809134 1853020195344644 150094635694716498 12157665465428043604

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.280225.1.
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.bc_np$2$(not in LMFDB)