Properties

Label 2.9.ak_br
Base Field $\F_{3^2}$
Dimension $2$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

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

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
$A(\F_{q^r})$ 25 5625 547600 44555625 3543225625 283875840000 22900866685225 1853050666055625 150078465576144400 12156915630659765625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 68 750 6788 60000 534158 4788000 43047428 387378750 3486569348

Decomposition

1.9.af 2

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}$2.3.a_af