Properties

Label 2.9.af_x
Base Field $\F_{3^2}$
Dimension $2$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^2}$
Dimension:  $2$
Weil polynomial:  $1 - 5 x + 23 x^{2} - 45 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.29396962675$, $\pm0.426020201528$
Angle rank:  $2$ (numerical)
Number field:  4.0.19525.1
Galois group:  $D_{4}$

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
$A(\F_{q^r})$ 55 8525 596695 43315525 3463262000 281995073525 22877094126095 1852927315071525 150088148121648055 12157788177524000000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 103 815 6603 58650 530623 4783035 43044563 387403745 3486819598

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.