# Properties

 Label 2.9.ak_br Base field $\F_{3^{2}}$ Dimension $2$ $p$-rank $2$ Ordinary Yes Supersingular No Simple No Geometrically simple No Primitive No Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 5 x + 9 x^{2} )^{2}$ Frobenius angles: $\pm0.186429498677$, $\pm0.186429498677$ 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:

• $y^2=ax^6+2ax^4+ax^3+2ax^2+a$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 25 5625 547600 44555625 3543225625 283875840000 22900866685225 1853050666055625 150078465576144400 12156915630659765625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 68 750 6788 60000 534158 4788000 43047428 387378750 3486569348

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{2}}$
 The isogeny class factors as 1.9.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$
All geometric endomorphisms are defined over $\F_{3^{2}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.

 Subfield Primitive Model $\F_{3}$ 2.3.a_af

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.9.a_ah $2$ 2.81.ao_id 2.9.k_br $2$ 2.81.ao_id 2.9.f_q $3$ 2.729.u_chy
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.9.a_ah $2$ 2.81.ao_id 2.9.k_br $2$ 2.81.ao_id 2.9.f_q $3$ 2.729.u_chy 2.9.a_h $4$ (not in LMFDB) 2.9.af_q $6$ (not in LMFDB)