Properties

Label 2.9.af_q
Base Field $\F_{3^2}$
Dimension $2$
$p$-rank $2$
Principally polarizable
Contains a Jacobian

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Invariants

Base field:  $\F_{3^2}$
Dimension:  $2$
Weil polynomial:  $1 - 5 x + 16 x^{2} - 45 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.146903834656$, $\pm0.519762832011$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-11})\)
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
$A(\F_{q^r})$ 48 7104 518400 42311424 3514999728 283875840000 22888823166768 1853004950436864 150110807438649600 12158040390598593984

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 89 710 6449 59525 534158 4785485 43046369 387462230 3486891929

Decomposition

This is a simple isogeny class.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^2}$.

SubfieldPrimitive Model
$\F_{3}$2.3.b_ac
$\F_{3}$2.3.ab_ac