# 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

## 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.

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

 $r$ 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.
 Twist Extension Degree Common 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