# Properties

 Label 2.81.abd_od 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: $1 - 29 x + 367 x^{2} - 2349 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.117314330344$, $\pm0.262733545783$ Angle rank: $2$ (numerical) Number field: 4.0.3244437.1 Galois group: $D_{4}$ Jacobians: 24

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 24 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4551 42356157 282691850175 1853523431686725 12158003602504751856 79766551113635315013525 523347626757290126395110711 3433683825425614248529597335525 22528399611462981457366098972523575 147808829519910504928186174392518522112

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 53 6455 531935 43058411 3486881378 282429919007 22876792180883 1853020191621971 150094635740211545 12157665467739893630

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.3244437.1.
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.bd_od $2$ (not in LMFDB)