Properties

Label 2.3.ab_d
Base Field $\F_{3}$
Dimension $2$
$p$-rank $1$
Principally polarizable
Contains a Jacobian

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
Weil polynomial:  $1 - x + 3 x^{2} - 3 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.268536328535$, $\pm0.622727850897$
Angle rank:  $2$ (numerical)
Number field:  4.0.11661.1
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
$A(\F_{q^r})$ 9 153 675 8109 70704 493425 4512951 43147989 384257925 3480899328

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 15 27 99 288 675 2061 6579 19521 58950

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.