# Properties

 Label 2.9.af_s Base field $\F_{3^{2}}$ 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: $\F_{3^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 5 x + 9 x^{2} )( 1 + 9 x^{2} )$ Frobenius angles: $\pm0.186429498677$, $\pm0.5$ 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, and hence is principally polarizable:

• $y^2=ax^6+(a+2)x^4+2ax^3+2x+a+2$
• $y^2=ax^6+(2a+1)x^4+2ax^3+(a+1)x+2a+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 50 7500 540200 42720000 3514951250 283929120000 22888831190450 1852470606720000 150086550606213800 12157702312129687500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 93 740 6513 59525 534258 4785485 43033953 387399620 3486794973

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{2}}$
 The isogeny class factors as 1.9.af $\times$ 1.9.a 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 $\overline{\F}_{3^{2}}$
 The base change of $A$ to $\F_{3^{4}}$ is 1.81.ah $\times$ 1.81.s. The endomorphism algebra for each factor is: 1.81.ah : $$\Q(\sqrt{-11})$$. 1.81.s : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{4}}$.

## 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 2.9.f_s $2$ 2.81.l_bk 2.9.al_bw $4$ (not in LMFDB) 2.9.ab_am $4$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.9.f_s $2$ 2.81.l_bk 2.9.al_bw $4$ (not in LMFDB) 2.9.ab_am $4$ (not in LMFDB) 2.9.b_am $4$ (not in LMFDB) 2.9.l_bw $4$ (not in LMFDB) 2.9.ai_bh $12$ (not in LMFDB) 2.9.ac_d $12$ (not in LMFDB) 2.9.c_d $12$ (not in LMFDB) 2.9.i_bh $12$ (not in LMFDB)