# Properties

 Label 3.3.ae_i_an 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: $1 - 4 x + 8 x^{2} - 13 x^{3} + 24 x^{4} - 36 x^{5} + 27 x^{6}$ Frobenius angles: $\pm0.102762435325$, $\pm0.278353759721$, $\pm0.643265352440$ Angle rank: $3$ (numerical) Number field: 6.0.10816643.1 Galois group: $A_4\times C_2$ 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:

• $2x^4+x^3y+x^2yz+x^2z^2+xy^2z+y^4+z^4=0$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 7 791 14623 655739 16743377 358570583 10309437859 289349423923 7647273092032 208557228479831

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 10 21 98 280 673 2156 6722 19740 59810

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 6.0.10816643.1.
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_i_n $2$ 3.9.a_i_at