Properties

Label 2.9.ai_bg
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:  $1 - 8 x + 32 x^{2} - 72 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.141826552031$, $\pm0.358173447969$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{8})\)
Galois group:  $C_2^2$

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
$A(\F_{q^r})$ 34 6596 561442 43507216 3481591874 282428906564 22898191896194 1854050612281344 150116130924799522 12157665465130939076

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 82 770 6630 58962 531442 4787442 43070654 387475970 3486784402

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.