Properties

Label 3.3.af_o_abc
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:  $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0975263560046$, $\pm0.304086723985$, $\pm0.527857038681$
Angle rank:  $3$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

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

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 6 972 20862 474336 15112416 395418348 10043458998 281036490624 7880965995582 210389494182912

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 13 29 73 254 745 2099 6529 20333 60328

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac $\times$ 2.3.ad_f and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3}$.

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.ab_c_ai$2$3.9.d_a_ae
3.3.b_c_i$2$3.9.d_a_ae
3.3.f_o_bc$2$3.9.d_a_ae