# Properties

 Label 2.81.abe_oq 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: $1 - 30 x + 380 x^{2} - 2430 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.0632554344274$, $\pm0.259213082638$ Angle rank: $2$ (numerical) Number field: 4.0.1696576.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4482 42139764 282381422562 1853185626758736 12157686905205490002 79766280290773257636564 523347411191392281383748882 3433683668045998059745580160000 22528399511777885815368791796115122 147808829472838339954981605752171670804

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 52 6422 531352 43050566 3486790552 282428960102 22876782757972 1853020106690558 150094635076063252 12157665463868084102

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.1696576.1.
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.be_oq $2$ (not in LMFDB)