# Properties

 Label 2.81.abc_nf 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: $1 - 28 x + 343 x^{2} - 2268 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.0378368453396$, $\pm0.309796496498$ Angle rank: $2$ (numerical) Number field: 4.0.3632400.1 Galois group: $D_{4}$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4609 42407409 282458416900 1852930711993449 12157261182123833809 79765944927862858592400 523347296814835850353056049 3433683716786073755345243799369 22528399580564708934075595577656900 147808829463530458027810961147215513809

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 54 6464 531498 43044644 3486668454 282427772678 22876777758294 1853020132993604 150094635534352938 12157665463102486304

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.3632400.1.
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.bc_nf $2$ (not in LMFDB)