Properties

Label 2.9.ae_h
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:  $1 - 4 x + 7 x^{2} - 36 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.0656128605321$, $\pm0.601053806135$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-5})\)
Galois group:  $C_2^2$

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
$A(\F_{q^r})$ 49 6321 470596 42028329 3500716849 281922769296 22857499894369 1853530045320009 150103193537593156 12157447877143900401

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 80 642 6404 59286 530486 4778934 43058564 387442578 3486722000

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.