Properties

Label 2.3.ac_g
Base Field $\F_{3}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $( 1 - 2 x + 3 x^{2} )( 1 + 3 x^{2} )$
Frobenius angles:  $\pm0.304086723985$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  2

This isogeny class is not simple.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

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
$A(\F_{q^r})$ 8 192 1064 6144 59048 536256 4599176 41779200 391199816 3544297152

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 18 38 78 242 738 2102 6366 19874 60018

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac $\times$ 1.3.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.c $\times$ 1.9.g. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.c_g$2$2.9.i_be
2.3.af_m$3$2.27.k_cc
2.3.b_a$3$2.27.k_cc
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.c_g$2$2.9.i_be
2.3.af_m$3$2.27.k_cc
2.3.b_a$3$2.27.k_cc
2.3.ab_a$6$2.729.i_abnm
2.3.f_m$6$2.729.i_abnm