# Properties

 Label 3.3.ae_l_aw 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: $1 - 4 x + 11 x^{2} - 22 x^{3} + 33 x^{4} - 36 x^{5} + 27 x^{6}$ Frobenius angles: $\pm0.132091637252$, $\pm0.376445424065$, $\pm0.544359499442$ Angle rank: $3$ (numerical) Number field: 6.0.5169344.1 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:

• $y^2=2x^8+2x^7+2x^4+2x^3+2x^2+2x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 10 1340 22030 455600 14700050 410330780 10709109190 293340793600 7859863377970 205846769956700

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 16 30 68 250 772 2240 6812 20280 59036

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 6.0.5169344.1.
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_l_w $2$ 3.9.g_l_i