Properties

Label 2.9.ae_p
Base Field $\F_{3^2}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^2}$
Dimension:  $2$
Weil polynomial:  $1 - 4 x + 15 x^{2} - 36 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.218106241758$, $\pm0.534324654653$
Angle rank:  $2$ (numerical)
Number field:  4.0.44688.2
Galois group:  $D_{4}$

This isogeny class is simple.

Newton polygon

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

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
$A(\F_{q^r})$ 57 7809 538308 43051017 3529298697 283488065424 22860713209881 1852260061640073 150092071627773636 12157513206547056609

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 96 738 6564 59766 533430 4779606 43029060 387413874 3486740736

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.