Properties

Label 2.3.b_ac
Base Field $\F_{3}$
Dimension $2$
$p$-rank $2$
Principally polarizable
Contains a Jacobian

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
Weil polynomial:  $1 + x - 2 x^{2} + 3 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.259881416005$, $\pm0.926548082672$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-11})\)
Galois group:  $V_4$

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
$A(\F_{q^r})$ 12 48 1296 7104 67332 518400 4606068 42311424 382124304 3514999728

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 5 44 89 275 710 2105 6449 19412 59525

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.