# Properties

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

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## Invariants

 Base field: $\F_{3^{4}}$ Dimension: $2$ L-polynomial: $1 - 30 x + 379 x^{2} - 2430 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.0439844089045$, $\pm0.263626170191$ Angle rank: $2$ (numerical) Number field: 4.0.69184.1 Galois group: $D_{4}$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4481 42125881 282333399104 1853095978864809 12157570794021386801 79766164027081734071296 523347315301061296337714801 3433683598435900758172770717129 22528399462262267650518070295833664 147808829433613660216210758339712522201

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 52 6420 531262 43048484 3486757252 282428548446 22876778566372 1853020069124804 150094634746167262 12157665460641750900

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.69184.1.
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_op $2$ (not in LMFDB)