# Properties

 Label 2.3.a_f Base field $\F_{3}$ Dimension $2$ $p$-rank $2$ Ordinary yes 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: $( 1 - x + 3 x^{2} )( 1 + x + 3 x^{2} )$ $1 + 5x^{2} + 9x^{4}$ Frobenius angles: $\pm0.406785250661$, $\pm0.593214749339$ Angle rank: $1$ (numerical) Jacobians: 2

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

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

## Point counts

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

• $y^2=2x^6+x^5+x^3+x+2$
• $y^2=x^6+2x^5+2x^3+2x+1$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $15$ $225$ $720$ $5625$ $58575$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $20$ $28$ $68$ $244$ $710$ $2188$ $6788$ $19684$ $58100$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ab $\times$ 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 $\F_{3^{2}}$ is 1.9.f 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$
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.ac_h$2$2.9.k_br
2.3.c_h$2$2.9.k_br
2.3.a_af$4$2.81.ao_id
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ac_h$2$2.9.k_br
2.3.c_h$2$2.9.k_br
2.3.a_af$4$2.81.ao_id
2.3.ab_ac$6$2.729.au_chy
2.3.b_ac$6$2.729.au_chy