Properties

Label 2.9.af_r
Base field $\F_{3^{2}}$
Dimension $2$
$p$-rank $2$
Ordinary Yes
Supersingular No
Simple Yes
Geometrically simple Yes
Primitive Yes
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 5 x + 17 x^{2} - 45 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.167045067981$, $\pm0.510218568904$
Angle rank:  $2$ (numerical)
Number field:  4.0.273325.1
Galois group:  $D_{4}$
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 49 7301 529249 42528325 3516460304 283968021701 22890166747889 1852751624552325 150098818271305729 12157846088656572416

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 91 725 6483 59550 534331 4785765 43040483 387431285 3486836206

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.273325.1.
All geometric endomorphisms are defined over $\F_{3^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.9.f_r$2$2.81.j_b