Properties

Label 2.81.abf_pl
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:  $1 - 31 x + 401 x^{2} - 2511 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.125544047245$, $\pm0.205363692800$
Angle rank:  $2$ (numerical)
Number field:  4.0.140125.1
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
$A(\F_{q^r})$ 4421 42021605 282413838101 1853502442275205 12158236620690620416 79766881318584619728005 523347878857065106261369621 3433683910568100629037696112005 22528399548352933962212049119890661 147808829389681210213442130821120000000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 51 6403 531411 43057923 3486948206 282431088163 22876803200771 1853020237569923 150094635319743171 12157665457028191198

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.140125.1.
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.bf_pl$2$(not in LMFDB)