# Properties

 Label 2.81.abc_nr 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: $1 - 28 x + 355 x^{2} - 2268 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.161515673273$, $\pm0.261305008072$ Angle rank: $2$ (numerical) Number field: 4.0.1911312.2 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4621 42573273 282995973364 1853829714911577 12158239439137856581 79766678366986982306832 523347645640407947554716949 3433683764090739377272547490153 22528399509587350399441439059372084 147808829413820480777577630855396809673

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 54 6488 532506 43065524 3486949014 282430369574 22876793006310 1853020158522020 150094635061468890 12157665459013709768

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.1911312.2.
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_nr $2$ (not in LMFDB)