Properties

Label 2.3.ac_b
Base Field $\F_{3}$
Dimension $2$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
Weil polynomial:  $1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.0292466093486$, $\pm0.637420057318$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-2}, \sqrt{-3})\)
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
$A(\F_{q^r})$ 3 57 324 5529 58323 467856 4600011 43264425 380016036 3458495577

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 8 8 68 242 638 2102 6596 19304 58568

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.