# Properties

 Label 2.81.abb_mu 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: $( 1 - 17 x + 81 x^{2} )( 1 - 10 x + 81 x^{2} )$ Frobenius angles: $\pm0.106600758076$, $\pm0.312505618912$ Angle rank: $2$ (numerical) Jacobians: 84

This isogeny class is not 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 84 curves, and hence is principally polarizable:

• $y^2=(a^3+2a^2+a+2)x^6+(2a^3+2a^2+2)x^5+(2a^3+2a^2+2a)x^4+(2a^3+a^2+2)x^3+(a^3+2a^2+a+2)x^2+(2a^2+a)x+2a^2+a+1$
• $y^2=2a^3x^6+(a^3+a^2+a)x^5+(a^2+2)x^4+(2a+2)x^3+(2a^3+2a+1)x^2+(a^2+a+1)x+a^2$
• $y^2=(a^2+2a+1)x^6+(2a^3+a^2+2a+2)x^5+(2a+1)x^4+(a^2+a+2)x^3+(2a^3+a^2+a)x^2+(2a^2+2a+1)x+2a^3+a^2+a+2$
• $y^2=(a^3+2a)x^6+(a^3+a^2)x^5+(2a^2+a+2)x^4+(a^2+2)x^3+(2a^2+a+1)x^2+(2a^3+a+2)x+2a^3$
• $y^2=(2a+2)x^6+(a^2+1)x^5+(a^3+2a^2)x^4+(a^3+2a^2+1)x^3+(a^3+2a^2+2a+1)x^2+(a^3+2a^2+a+1)x+2a^3+a^2+2a+1$
• $y^2=a^3x^6+(2a^3+2a+2)x^5+(2a^3+2a^2+2a+1)x^4+(2a^3+2a^2+2a)x^3+(2a^3+a^2+2a)x^2+(2a^3+2a+1)x+a^3+2a^2+2a+2$
• $y^2=(a^3+2a^2+a)x^6+(2a^3+a^2+2)x^5+x^4+(a^3+2a^2+2)x^3+(2a^3+1)x^2+(a^3+a+2)x+2a^3+a+2$
• $y^2=(2a^2+a+1)x^6+(a^3+a^2+a+1)x^5+(a^3+2a+2)x^4+a^3x^3+(a^3+a)x^2+(2a^3+1)x+2a^2+1$
• $y^2=(2a^2+2a+2)x^6+(a^3+a^2+2a)x^5+(2a+1)x^4+(a^2+2a)x^3+(2a+1)x^2+(2a^3+a^2+2a+2)x+a^3+a$
• $y^2=(a^3+2a^2+1)x^6+(2a^2+a+2)x^5+(a^3+1)x^4+(2a^3+a^2+a+1)x^3+(2a^3+a^2+2)x^2+(a^3+a^2+2)x+a^3+2a^2+a+1$
• $y^2=(2a^3+a^2+2a+1)x^6+(2a^3+2a^2+a)x^5+x^4+(a^3+2a^2+2a+1)x^3+a^3x^2+(2a^3+2a^2+1)x+a^3+2a^2+a+1$
• $y^2=(2a^2+a+1)x^6+ax^5+(2a^3+a^2+a)x^4+(2a^3+2a^2+a+1)x^3+(a^3+2a+1)x^2+(a^3+2a+2)x+a^3+a+2$
• $y^2=(2a^3+a^2+a)x^6+(a^3+2a+2)x^5+(a^2+a)x^4+(2a^3+a^2+a)x^3+(a^3+2a+1)x^2+2a^2x+2a^3+2a^2+a$
• $y^2=(a^3+a^2+2a)x^6+(a^3+a^2+2a+1)x^5+(2a^3+a^2+2a+2)x^4+(2a^3+a^2+2a)x^3+(2a^3+2a^2+2a+1)x^2+(2a^3+a)x+a^3+2a^2+a+1$
• $y^2=(2a^3+2a+1)x^6+(2a^3+2)x^5+(2a^3+a+2)x^4+(2a^3+2a)x^3+(a^2+2a)x^2+(a^3+2)x+a^3+a^2+2a$
• $y^2=(2a^2+a+1)x^6+(2a^2+1)x^5+(a^3+2a^2+a+2)x^4+x^3+(2a^3+2a)x^2+(a^3+2a^2+a)x+a^3+2a$
• $y^2=(2a^3+a^2)x^6+(a^3+2a^2+a+2)x^5+(a^3+2a+1)x^4+(2a^3+2a^2+2a)x^3+(a^3+a^2+2a+2)x^2+(2a^3+a^2+2a+1)x+a^3+2a+1$
• $y^2=(2a^3+2a^2+a+1)x^6+(2a^3+a)x^4+(a^3+a^2+2a+1)x^3+(a^3+2a^2+a+1)x^2+(2a^2+2a+2)x+2a^3+2a^2+a+1$
• $y^2=(a^3+2a+1)x^6+(2a^3+2a^2+2)x^4+(2a^3+2a^2+2a+2)x^3+(2a^3+a+1)x^2+(2a^3+2a^2+2a)x+2a^3+2a^2+2a$
• $y^2=(2a^2+2)x^6+(a^3+a)x^5+(a^3+a^2)x^4+(a^3+2a^2+2a+2)x^3+(a^3+2a^2+2a)x^2+(a^2+a)x+2a^2+a+2$
• and 64 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4680 42625440 282773855520 1853290193040000 12157627714334037000 79766293202936297717760 523347603971378905253288520 3433683963092832019248575040000 22528399757003972151023316279682080 147808829575634782192479178981954596000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 55 6497 532090 43052993 3486773575 282429005822 22876791184855 1853020265915393 150094636709873050 12157665472323362177

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.ar $\times$ 1.81.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ah_ai $2$ (not in LMFDB) 2.81.h_ai $2$ (not in LMFDB) 2.81.bb_mu $2$ (not in LMFDB)