Properties

Label 3.3.ae_m_ax
Base Field $\F_{3}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $1 - 4 x + 12 x^{2} - 23 x^{3} + 36 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:  $\pm0.200090550351$, $\pm0.363029791168$, $\pm0.522713769744$
Angle rank:  $3$ (numerical)
Number field:  6.0.1178891.1
Galois group:  $A_4\times C_2$
Jacobians:  0

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 is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 13 1807 30589 529451 15285803 412431487 10452303329 281037355859 7675376218432 204804523339247

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 18 39 82 260 777 2184 6530 19812 58738

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is 6.0.1178891.1.
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_m_x$2$3.9.i_bg_dx