Properties

Label 2.3.a_af
Base Field $\F_{3}$
Dimension $2$
$p$-rank $2$
Principally polarizable
Does not contain a Jacobian

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
Weil polynomial:  $1 - 5 x^{2} + 9 x^{4}$
Frobenius angles:  $\pm0.0932147493387$, $\pm0.906785250661$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{11})\)
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 is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 5 25 740 5625 59525 547600 4785485 44555625 387399620 3543225625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 0 28 68 244 750 2188 6788 19684 60000

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.