Properties

Label 3.3.ae_j_ar
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:  $1 - 4 x + 9 x^{2} - 17 x^{3} + 27 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:  $\pm0.0653366913680$, $\pm0.328985474983$, $\pm0.609104440316$
Angle rank:  $3$ (numerical)
Number field:  6.0.10338167.1
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
$A(\F_{q^r})$ 7 847 14308 502271 14422352 339328528 9863032277 292370442287 7771038924292 206089871896832

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 12 21 76 245 633 2058 6788 20055 59107

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is 6.0.10338167.1.
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_j_r$2$3.9.c_ab_abl