Properties

Label 2.3.a_d
Base Field $\F_{3}$
Dimension $2$
$p$-rank $0$
Principally polarizable
Contains a Jacobian

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
Weil polynomial:  $1 + 3 x^{2} + 9 x^{4}$
Frobenius angles:  $\pm0.333333333333$, $\pm0.666666666667$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\zeta_{12})\)
Galois group:  $V_4$

This isogeny class is simple.

Newton polygon

This isogeny class is supersingular.

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

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
$A(\F_{q^r})$ 13 169 676 8281 59293 456976 4785157 44129449 387381124 3515659849

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 16 28 100 244 622 2188 6724 19684 59536

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.