Properties

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

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 + x + 3 x^{2} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.593214749339$
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
$A(\F_{q^r})$ 5 105 560 6825 74525 564480 4776245 44342025 390136880 3444620025

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 12 20 84 302 774 2186 6756 19820 58332

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.b 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 $\F_{3^{6}}$ is 1.729.ak $\times$ 1.729.cc. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field

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.ae_j$2$2.9.c_d
2.3.c_d$2$2.9.c_d
2.3.e_j$2$2.9.c_d
2.3.b_g$3$2.27.ai_cc
2.3.e_j$3$2.27.ai_cc
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.ae_j$2$2.9.c_d
2.3.c_d$2$2.9.c_d
2.3.e_j$2$2.9.c_d
2.3.b_g$3$2.27.ai_cc
2.3.e_j$3$2.27.ai_cc
2.3.ab_g$6$2.729.bs_bji