Properties

Label 3.3.ae_h_ak
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:  $1 - 4 x + 7 x^{2} - 10 x^{3} + 21 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:  $\pm0.0823229705598$, $\pm0.256885878434$, $\pm0.668023839470$
Angle rank:  $3$ (numerical)
Number field:  6.0.7181504.1
Galois group:  $S_4\times C_2$
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 6 636 12978 663984 16071726 356298012 10446099306 282177264384 7531434580446 208624921594716

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 8 18 100 270 668 2184 6556 19440 59828

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is 6.0.7181504.1.
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_h_k$2$3.9.ac_l_abo