# Properties

 Label 3.3.ae_n_az Base Field $\F_{3}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - x + 3 x^{2} )( 1 - 3 x + 7 x^{2} - 9 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.227267020856$, $\pm0.406785250661$, $\pm0.464830336654$ 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 $A(\F_{q^r})$ 15 2175 37980 489375 13206000 418539600 11061938805 277049379375 7377948958140 203895357600000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 20 45 76 225 785 2310 6436 19035 58475

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ab $\times$ 2.3.ad_h 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.ac_h_al $2$ 3.9.k_bv_fz 3.3.c_h_l $2$ 3.9.k_bv_fz 3.3.e_n_z $2$ 3.9.k_bv_fz