Properties

Label 2.3.ad_f
Base Field $\F_{3}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.0975263560046$, $\pm0.527857038681$
Angle rank:  $2$ (numerical)
Number field:  4.0.2197.1
Galois group:  $C_4$
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 3 81 549 4941 62448 578097 4778049 43050933 396546543 3534057216

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1 11 19 59 256 791 2185 6563 20143 59846

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is 4.0.2197.1.
All geometric endomorphisms are defined over $\F_{3}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.d_f$2$2.9.b_al