# Properties

 Label 2.3.a_f Base Field $\F_{3}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - x + 3 x^{2} )( 1 + x + 3 x^{2} )$ 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, 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 6 7 8 9 10 $A(\F_{q^r})$ 15 225 720 5625 58575 518400 4780455 44555625 387441360 3431030625

 $r$ 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.
 Twist Extension Degree Common 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.
 Twist Extension Degree Common 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