Properties

Label 3.3.af_p_abf
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:  $1 - 5 x + 15 x^{2} - 31 x^{3} + 45 x^{4} - 45 x^{5} + 27 x^{6}$
Frobenius angles:  $\pm0.113296540390$, $\pm0.351823865540$, $\pm0.481790494592$
Angle rank:  $3$ (numerical)
Number field:  6.0.400967.1
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
$A(\F_{q^r})$ 7 1183 26173 445991 13038697 402514567 10956704192 288660092903 7775839770139 208787598926543

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 15 35 67 219 759 2288 6707 20069 59875

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is 6.0.400967.1.
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.f_p_bf$2$3.9.f_f_ah