# Properties

 Label 2.81.abb_mq Base Field $\F_{3^{4}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3^{4}}$ Dimension: $2$ L-polynomial: $1 - 27 x + 328 x^{2} - 2187 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.0728119855854$, $\pm0.323673068911$ Angle rank: $2$ (numerical) Number field: 4.0.257725.1 Galois group: $D_{4}$ Jacobians: 32

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 32 curves, and hence is principally polarizable:

• $y^2=(2a^3+2)x^6+(2a^2+2a)x^5+a^2x^4+(a^3+2a^2+2a+2)x^3+(2a^2+a+2)x^2+(a^3+a^2+a)x+a^3+2a^2+2a+2$
• $y^2=(2a^3+2a)x^6+(2a^3+a^2)x^5+(2a^3+a)x^4+(2a^3+a+2)x^3+(2a^2+a)x^2+(2a^3+a+2)x+a+2$
• $y^2=(a^2+a+1)x^6+(2a^2+a+2)x^5+(a^3+a^2+2)x^4+x^3+(2a^3+2a^2+2a)x^2+(a^2+a+2)x+2a^3+2a^2+a+2$
• $y^2=(a^3+a+2)x^6+(a^3+2a^2+a+1)x^5+(a^2+a)x^4+(a^3+2a^2+a+2)x^3+(a^3+a^2+2a+1)x^2+2ax+2a^2+2a+1$
• $y^2=(a^3+2a^2+1)x^6+(a^3+2a+2)x^5+(a^3+a^2+a+1)x^4+x^3+(a^3+2a)x^2+(a^3+2a+1)x+a^3+2a^2$
• $y^2=2a^3x^6+(2a^3+a^2+1)x^5+2x^4+a^2x^3+(2a^2+2a+1)x^2+(2a+1)x+a^2+2a+1$
• $y^2=2ax^6+(2a^3+2a+1)x^5+(a^3+2)x^4+(a^2+2)x^3+(a^3+a^2+2a+1)x^2+(2a^3+a^2+2a+2)x+2a+1$
• $y^2=(a^3+a+2)x^6+(a^3+2a^2+1)x^5+(a^3+2a^2+a)x^4+(2a^3+2a^2+2)x^3+(a+2)x^2+a^3x+a^3+2a^2+2a+1$
• $y^2=(a^3+2a^2)x^6+(a^3+2a^2+a+2)x^5+(a^3+a^2+2a)x^4+(a^3+a^2+2a+2)x^3+(2a^3+2a+1)x^2+(a^2+1)x+a^3+2a^2+a+2$
• $y^2=(a^3+2a^2+a+1)x^6+(2a^3+1)x^5+(2a^2+2a)x^4+(2a^3+2a+2)x^3+2a^2x^2+(2a^3+a^2+2a+1)x+a^3+2a+1$
• $y^2=(a^2+a)x^6+(a^3+a^2+a)x^5+(a^2+a+1)x^4+(2a^3+2a^2)x^3+(2a^3+2a^2+2a+1)x^2+(a^3+a^2+a)x+a^2+1$
• $y^2=(2a^3+a^2+a)x^6+(a^2+a+1)x^5+(a+2)x^4+(a^3+2a^2+1)x^3+ax^2+(2a^3+2a+1)x+a^3+2a^2+2a+2$
• $y^2=(a^3+2)x^6+(a+1)x^5+(2a^2+2)x^4+(a^3+a^2+2)x^3+(2a^3+a^2+1)x^2+(2a^3+a^2+a)x+a^3+2a^2+a$
• $y^2=(2a^3+2a)x^5+(a^3+2a^2+2)x^4+2ax^3+(a^3+a+2)x^2+(a^2+2a+1)x+a^3+2a^2+1$
• $y^2=(a^3+2a^2+2a)x^6+(2a^3+a^2+a+1)x^5+(a^3+2a^2+a+2)x^4+(a^3+2a^2+1)x^3+(a^2+a+2)x^2+(2a^2+2a)x+2a^2+a+1$
• $y^2=(2a^3+2a^2+a)x^6+(2a^3+2a^2+a)x^5+(2a^3+2a)x^4+(a^3+a+2)x^3+(2a^3+2a^2+1)x^2+(2a^3+a+1)x+a^2+2$
• $y^2=(a^3+2a^2)x^6+2ax^5+(2a^3+a^2+2a+1)x^4+(a^3+2a^2)x^3+(a^3+a^2+2a+1)x^2+(2a^3+a^2+a+2)x+a+2$
• $y^2=(a^3+2a^2+2a)x^6+(a+1)x^5+(a^2+a)x^4+(a^3+a+1)x^3+(2a+2)x^2+(a^3+2a+1)x+a^3+a+1$
• $y^2=(2a^3+a^2+2a+2)x^6+(2a^3+a+2)x^5+(a^2+a+1)x^4+(a^3+1)x^3+(2a^3+a)x^2+(2a^3+2a)x+2a^3+a^2+1$
• $y^2=(2a^2+1)x^6+(a^3+2a^2+2a)x^5+(a^2+a)x^4+(2a+1)x^3+(a^3+2a^2+a+1)x^2+(a^3+1)x+a^3+a^2+a$
• and 12 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4676 42570304 282601148816 1853015347540224 12157345287178245476 79766088706538539110400 523347492765511920526165796 3433683905389072304141132104704 22528399711446719288002649309272976 147808829532433331943284638217361822784

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 55 6489 531766 43046609 3486692575 282428281758 22876786323775 1853020234775009 150094636406349526 12157665468769929129

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.257725.1.
All geometric endomorphisms are defined over $\F_{3^{4}}$.

## 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 2.81.bb_mq $2$ (not in LMFDB)