Properties

Label 3.3.ae_m_aw
Base field $\F_{3}$
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:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x + 3 x^{2} )( 1 - 2 x + 5 x^{2} - 6 x^{3} + 9 x^{4} )$
  $1 - 4 x + 12 x^{2} - 22 x^{3} + 36 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:  $\pm0.254551732336$, $\pm0.304086723985$, $\pm0.538152604671$
Angle rank:  $3$ (numerical)
Jacobians:  $0$
Isomorphism classes:  1

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14$ $1932$ $32984$ $633696$ $16738414$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $18$ $42$ $98$ $280$ $720$ $1960$ $6274$ $19950$ $59778$

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 $\F_{3}$.

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac $\times$ 2.3.ac_f 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
3.3.a_e_ac$2$3.9.i_bo_fq
3.3.a_e_c$2$3.9.i_bo_fq
3.3.e_m_w$2$3.9.i_bo_fq