Properties

Label 3.3.af_q_abh
Base Field $\F_{3}$
Dimension $3$
$p$-rank $2$
Does not contain a Jacobian

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
Weil polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - x + 3 x^{2} )^{2}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.406785250661$, $\pm0.406785250661$
Angle rank:  $1$ (numerical)

Newton polygon

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

Point counts

This isogeny class does not contain a Jacobian, and it is unknown whether it 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 1575 36288 511875 12294999 406425600 11702233629 295983016875 7521734799936 201768617964375

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 17 44 77 209 764 2435 6869 19412 57857

Decomposition

1.3.ad $\times$ 1.3.ab 2

Base change

This is a primitive isogeny class.