Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
Weil polynomial:  $( 1 - x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{2}$
Frobenius angles:  $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.406785250661$
Angle rank:  $2$ (numerical)

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

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
$A(\F_{q^r})$ 12 2160 51984 691200 12474132 336856320 10034195484 284453683200 7720988424048 207593257330800

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 19 56 103 209 628 2099 6607 19928 59539

Decomposition

1.3.ac 2 $\times$ 1.3.ab

Base change

This is a primitive isogeny class.