Properties

Label 2.9.ae_q
Base Field $\F_{3^2}$
Dimension $2$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^2}$
Dimension:  $2$
Weil polynomial:  $1 - 4 x + 16 x^{2} - 36 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.234075997255$, $\pm0.523868533114$
Angle rank:  $2$ (numerical)
Number field:  4.0.334080.4
Galois group:  $D_{4}$

This isogeny class is simple.

Newton polygon

This isogeny class is ordinary.

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

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
$A(\F_{q^r})$ 58 8004 547114 43061520 3522170698 283358000196 22857501996058 1852033947156480 150089761407718714 12157769283627262404

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 98 750 6566 59646 533186 4778934 43023806 387407910 3486814178

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.