# Properties

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

## 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:

• $y^2=2x^5+x^4+2x^3+x^2+2x$
• $y^2=2x^6+2x^5+x^4+x^3+x^2+2x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 8 192 1064 6144 59048 536256 4599176 41779200 391199816 3544297152

 $r$ 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: 1.9.c : $$\Q(\sqrt{-2})$$. 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
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.
 Twist Extension Degree Common 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.
 Twist Extension Degree Common 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