# Properties

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

## Invariants

 Base field: $\F_{3^{2}}$ Dimension: $2$ L-polynomial: $1 - 5 x + 17 x^{2} - 45 x^{3} + 81 x^{4}$ Frobenius angles: $\pm0.167045067981$, $\pm0.510218568904$ Angle rank: $2$ (numerical) Number field: 4.0.273325.1 Galois group: $D_{4}$ Jacobians: 4

This isogeny class is simple and geometrically 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 4 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 49 7301 529249 42528325 3516460304 283968021701 22890166747889 1852751624552325 150098818271305729 12157846088656572416

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 91 725 6483 59550 534331 4785765 43040483 387431285 3486836206

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.273325.1.
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.f_r $2$ 2.81.j_b