Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
Weil polynomial:  $1 - x^{2} + 9 x^{4}$
Frobenius angles:  $\pm0.223349810481$, $\pm0.776650189519$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-5}, \sqrt{7})\)
Galois group:  $V_4$

This isogeny class is simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 9 81 756 9801 58689 571536 4787001 41409225 387381204 3444398721

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 8 28 116 244 782 2188 6308 19684 58328

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.