Properties

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

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $2$
Weil polynomial:  $1 - x + 5 x^{2} - 3 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.345303779071$, $\pm0.557095674046$
Angle rank:  $2$ (numerical)
Number field:  4.0.2725.1
Galois group:  $D_{4}$

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
$A(\F_{q^r})$ 11 209 869 6061 60016 511841 4542241 43693749 398261831 3484769024

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 19 33 75 248 703 2075 6659 20229 59014

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.