Properties

Label 2.81.abd_oh
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:  $1 - 29 x + 371 x^{2} - 2349 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.165617608067$, $\pm0.233191665838$
Angle rank:  $2$ (numerical)
Number field:  4.0.290125.1
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4555 42411605 282877529755 1853848619174005 12158375720438750000 79766836734008238973805 523347736634102091930211355 3433683758164898343319027264005 22528399437637839403187723911958755 147808829331679663039543606140020000000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 53 6463 532283 43065963 3486988098 282430930303 22876796983863 1853020155324083 150094634582107913 12157665452257411198

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.290125.1.
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.bd_oh$2$(not in LMFDB)