# Properties

 Label 2.3.ac_g 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

# Related objects

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - 2 x + 3 x^{2} )( 1 + 3 x^{2} )$ $1 - 2 x + 6 x^{2} - 6 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.304086723985$, $\pm0.5$ Angle rank: $1$ (numerical) Jacobians: 2

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

## Newton polygon

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

## Point counts

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

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $8$ $192$ $1064$ $6144$ $59048$

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

## 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.
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