Properties

Label 2.9.af_p
Base Field $\F_{3^2}$
Dimension $2$
$p$-rank $1$
Principally polarizable
Contains a Jacobian

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Invariants

Base field:  $\F_{3^2}$
Dimension:  $2$
Weil polynomial:  $1 - 5 x + 15 x^{2} - 45 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.125262572547$, $\pm0.528760283814$
Angle rank:  $2$ (numerical)
Number field:  4.0.28749.1
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
$A(\F_{q^r})$ 47 6909 507647 42068901 3510562352 283646238309 22883798598407 1853161382113125 150119763264599807 12158209061771291904

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 87 695 6411 59450 533727 4784435 43050003 387485345 3486940302

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.