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

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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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.
TwistExtension DegreeCommon base change
3.3.e_k_u$2$3.9.e_a_abi