# Properties

 Label 3.3.ae_k_au Base Field $\F_{3}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $1 - 4 x + 10 x^{2} - 20 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6}$ Frobenius angles: $\pm0.0844416807585$, $\pm0.360432408976$, $\pm0.575465777728$ Angle rank: $3$ (numerical) Number field: 6.0.2296688.1 Galois group: $S_4\times C_2$ Jacobians: 1

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 contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2=2x^8+x^7+x^6+x^5+2x^3+2x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 8 1024 16736 446464 14108968 368459776 10181082136 295125204992 7879007401376 205627034297344

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 14 24 66 240 692 2128 6850 20328 58974

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 6.0.2296688.1.
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_k_u $2$ 3.9.e_a_abi