Properties

Label 3.3.ae_h_ak
Base field $\F_{3}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
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:  $2$
Isomorphism classes:  2

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6$ $636$ $12978$ $663984$ $16071726$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which 1 is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3}$.

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

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