# Properties

 Label 3.3.ae_l_ax 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: $( 1 - x + 3 x^{2} )( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0975263560046$, $\pm0.406785250661$, $\pm0.527857038681$ 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 $A(\F_{q^r})$ 9 1215 19764 370575 13301424 416229840 10850949279 287364977775 7751691822564 207007401427200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 16 27 52 225 781 2268 6676 20007 59371

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ab $\times$ 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.ac_f_an $2$ 3.9.g_d_abl 3.3.c_f_n $2$ 3.9.g_d_abl 3.3.e_l_x $2$ 3.9.g_d_abl