Properties

Label 2.3.a_e
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:  $1 + 4 x^{2} + 9 x^{4}$
Frobenius angles:  $\pm0.366139763599$, $\pm0.633860236401$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{2}, \sqrt{-5})\)
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
$A(\F_{q^r})$ 14 196 686 7056 58814 470596 4787006 45158400 387431534 3459086596

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 18 28 86 244 642 2188 6878 19684 58578

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.