# Properties

 Label 2.9.ag_x Base field $\F_{3^{2}}$ 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: $\F_{3^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 5 x + 9 x^{2} )( 1 - x + 9 x^{2} )$ Frobenius angles: $\pm0.186429498677$, $\pm0.446699620962$ Angle rank: $2$ (numerical) Jacobians: 12

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 12 curves, and hence is principally polarizable:

• $y^2=(a+1)x^6+(a+1)x^5+2ax^4+2ax^3+(2a+2)x^2+x+2a+1$
• $y^2=(2a+1)x^6+(a+2)x^5+(a+1)x^4+2x^3+(2a+1)x^2+a+2$
• $y^2=(2a+2)x^6+(a+2)x^5+(2a+2)x^4+(a+1)x^3+(2a+2)x^2+(a+2)x+2a+2$
• $y^2=x^6+x^5+ax^4+(2a+1)x^3+ax^2+x+a+2$
• $y^2=(2a+1)x^6+(a+2)x^5+ax^4+(a+1)x^3+(a+2)x^2+(a+1)x+2a$
• $y^2=x^6+(2a+1)x^5+x^4+2x^3+x^2+(2a+1)x+1$
• $y^2=(2a+1)x^6+(2a+1)x^5+2x^4+x^3+x^2+(2a+1)x+a+2$
• $y^2=(2a+1)x^6+x^5+(2a+2)x^4+x^2+2x+a$
• $y^2=(a+2)x^6+(a+1)x^5+(2a+2)x^3+(a+1)x^2+ax+2a+1$
• $y^2=(2a+1)x^6+x^5+x^4+(2a+2)x^3+(a+1)x^2+(2a+1)x+a$
• $y^2=ax^6+x^5+(a+1)x^4+x^2+2x+2a+1$
• $y^2=(a+2)x^6+(a+2)x^5+(a+1)x^4+(2a+2)x^3+(2a+2)x^2+(a+2)x+2a+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 45 7425 559440 42953625 3493462725 283568947200 22908121480485 1852906027883625 150071331224742480 12157247924912660625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 92 766 6548 59164 533582 4789516 43044068 387360334 3486664652

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{2}}$
 The isogeny class factors as 1.9.af $\times$ 1.9.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.9.ae_n $2$ 2.81.k_br 2.9.e_n $2$ 2.81.k_br 2.9.g_x $2$ 2.81.k_br