# Properties

 Label 2.3.ab_a Base field $\F_{3}$ Dimension $2$ $p$-rank $1$ Ordinary no Supersingular no Simple no Geometrically simple no Primitive yes Principally polarizable yes Contains a Jacobian yes

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 + 2 x + 3 x^{2} )$ $1 - x - 3 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.166666666667$, $\pm0.695913276015$ Angle rank: $1$ (numerical) Jacobians: 1

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 Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+x^4+2x$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6$ $84$ $504$ $8736$ $66666$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $9$ $18$ $105$ $273$ $738$ $2355$ $6609$ $19494$ $59289$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.c 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.abu $\times$ 1.729.cc. The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 1.9.c. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak $\times$ 1.27.a. The endomorphism algebra for each factor is:

## 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.af_m$2$2.9.ab_m
2.3.b_a$2$2.9.ab_m
2.3.f_m$2$2.9.ab_m
2.3.c_g$3$2.27.ak_cc
2.3.f_m$3$2.27.ak_cc
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.af_m$2$2.9.ab_m
2.3.b_a$2$2.9.ab_m
2.3.f_m$2$2.9.ab_m
2.3.c_g$3$2.27.ak_cc
2.3.f_m$3$2.27.ak_cc
2.3.af_m$6$2.729.i_abnm
2.3.ac_g$6$2.729.i_abnm
2.3.c_g$6$2.729.i_abnm