# Properties

 Label 2.81.abi_rj 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: $( 1 - 17 x + 81 x^{2} )^{2}$ Frobenius angles: $\pm0.106600758076$, $\pm0.106600758076$ Angle rank: $1$ (numerical) Jacobians: 5

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 5 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4225 41409225 281600035600 1852761401001225 12157750704111390625 79766698031302547865600 523347938799711239069860225 3433684105848095743930383465225 22528399775649284393996212438243600 147808829580277804007459243690942015625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 48 6308 529878 43040708 3486808848 282430439198 22876805821008 1853020342954628 150094636834096758 12157665472705262948

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.ar 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-35})$$$)$
All geometric endomorphisms are defined over $\F_{3^{4}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.a_aex $2$ (not in LMFDB) 2.81.bi_rj $2$ (not in LMFDB) 2.81.r_ia $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.a_aex $2$ (not in LMFDB) 2.81.bi_rj $2$ (not in LMFDB) 2.81.r_ia $3$ (not in LMFDB) 2.81.a_ex $4$ (not in LMFDB) 2.81.ar_ia $6$ (not in LMFDB)