Properties

Label 3.9.am_cu_aki
Base field $\F_{3^{2}}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{3^{2}}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x + 9 x^{2} )( 1 - 8 x + 31 x^{2} - 72 x^{3} + 81 x^{4} )$
  $1 - 12 x + 72 x^{2} - 268 x^{3} + 648 x^{4} - 972 x^{5} + 729 x^{6}$
Frobenius angles:  $\pm0.0954872438962$, $\pm0.267720472801$, $\pm0.376614839446$
Angle rank:  $3$ (numerical)
Isomorphism classes:  20

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $198$ $534996$ $420216984$ $286629456960$ $205017018156918$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $82$ $790$ $6658$ $58798$ $530296$ $4782958$ $43057666$ $387458422$ $3486876082$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3^{2}}$
The isogeny class factors as 1.9.ae $\times$ 2.9.ai_bf and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
3.9.ae_i_au$2$(not in LMFDB)
3.9.e_i_u$2$(not in LMFDB)
3.9.m_cu_ki$2$(not in LMFDB)