# Properties

 Label 3.3.ae_k_av Base Field $\F_{3}$ Dimension $3$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $1 - 4 x + 10 x^{2} - 21 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6}$ Frobenius angles: $\pm0.0145064862012$, $\pm0.383559653096$, $\pm0.564732805964$ Angle rank: $2$ (numerical) Number field: 6.0.309123.1 Galois group: $D_{6}$ Jacobians: 0

This isogeny class is simple and geometrically simple.

## Newton polygon

 $p$-rank: $2$ Slopes: $[0, 0, 1/2, 1/2, 1, 1]$

## 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 $A(\F_{q^r})$ 7 903 14539 358491 12358717 354475359 9850388611 283090663443 7647092485312 202171542731823

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 14 21 50 210 665 2058 6578 19740 57974

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 6.0.309123.1.
All geometric endomorphisms are defined over $\F_{3}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.e_k_v $2$ 3.9.e_ai_acx 3.3.c_ac_aj $3$ (not in LMFDB) 3.3.c_e_d $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.e_k_v $2$ 3.9.e_ai_acx 3.3.c_ac_aj $3$ (not in LMFDB) 3.3.c_e_d $3$ (not in LMFDB) 3.3.ac_ac_j $6$ (not in LMFDB) 3.3.ac_e_ad $6$ (not in LMFDB)