# Properties

 Label 2.3.ac_d 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: $( 1 - 3 x + 3 x^{2} )( 1 + x + 3 x^{2} )$ $1 - 2x + 3x^{2} - 6x^{3} + 9x^{4}$ 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 (of which all are hyperelliptic), and hence is principally polarizable:

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $5$ $105$ $560$ $6825$ $74525$

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

## 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: 1.729.ak : $$\Q(\sqrt{-11})$$. 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.f. 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.ai $\times$ 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.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