# Properties

 Label 2.3.ac_h 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: $( 1 - x + 3 x^{2} )^{2}$ Frobenius angles: $\pm0.406785250661$, $\pm0.406785250661$ Angle rank: $1$ (numerical) Jacobians: 1

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 1 curves, and hence is principally polarizable:

• $y^2=2x^6+x^4+2x^3+x^2+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 9 225 1296 5625 45369 518400 5157441 44555625 382124304 3431030625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 20 44 68 182 710 2354 6788 19412 58100

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$
All geometric endomorphisms are defined over $\F_{3}$.

## 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.a_f $2$ 2.9.k_br 2.3.c_h $2$ 2.9.k_br 2.3.b_ac $3$ 2.27.q_eo
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.a_f $2$ 2.9.k_br 2.3.c_h $2$ 2.9.k_br 2.3.b_ac $3$ 2.27.q_eo 2.3.a_af $4$ 2.81.ao_id 2.3.ab_ac $6$ 2.729.au_chy