# Properties

 Label 3.3.ae_m_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: $1 - 4 x + 12 x^{2} - 23 x^{3} + 36 x^{4} - 36 x^{5} + 27 x^{6}$ Frobenius angles: $\pm0.200090550351$, $\pm0.363029791168$, $\pm0.522713769744$ Angle rank: $3$ (numerical) Number field: 6.0.1178891.1 Galois group: $A_4\times C_2$ Jacobians: 0

This isogeny class is simple and geometrically 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 $A(\F_{q^r})$ 13 1807 30589 529451 15285803 412431487 10452303329 281037355859 7675376218432 204804523339247

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 18 39 82 260 777 2184 6530 19812 58738

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 6.0.1178891.1.
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.e_m_x $2$ 3.9.i_bg_dx