Properties

Label 3.9.an_de_alx
Base Field $\F_{3^2}$
Dimension $3$
$p$-rank $2$
Does not contain a Jacobian

Learn more about

Invariants

Base field:  $\F_{3^2}$
Dimension:  $3$
Weil polynomial:  $( 1 - 3 x + 9 x^{2} )( 1 - 5 x + 9 x^{2} )^{2}$
Frobenius angles:  $\pm0.186429498677$, $\pm0.186429498677$, $\pm0.333333333333$
Angle rank:  $1$ (numerical)

Newton polygon

$p$-rank:  $2$
Slopes:  $[0, 0, 1/2, 1/2, 1, 1]$

Point counts

This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 175 511875 429318400 295983016875 208366469329375 150449653186560000 109484074133990031175 79779914739031316866875 58149380660833866259206400 42389261651125538578726171875

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 77 804 6869 59757 532700 4785813 43053989 387418116 3486628397

Decomposition

1.9.af 2 $\times$ 1.9.ad

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}$3.3.ad_ac_p
$\F_{3}$3.3.d_ac_ap