# Properties

 Label 2.81.abg_qc 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: $( 1 - 16 x + 81 x^{2} )^{2}$ Frobenius angles: $\pm0.151478024726$, $\pm0.151478024726$ Angle rank: $1$ (numerical) Jacobians: 16

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 16 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4356 41835024 282209562756 1853389289816064 12158261183469270276 79767019018955247210000 523348062861561142272444036 3433684071279087378501042438144 22528399641786318946630573420927236 147808829406475270192486728160410157584

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 50 6374 531026 43055294 3486955250 282431575718 22876811244050 1853020324299134 150094635942239666 12157665458409546854

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.aq 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-17})$$$)$
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_adq $2$ (not in LMFDB) 2.81.bg_qc $2$ (not in LMFDB) 2.81.q_gt $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.a_adq $2$ (not in LMFDB) 2.81.bg_qc $2$ (not in LMFDB) 2.81.q_gt $3$ (not in LMFDB) 2.81.a_dq $4$ (not in LMFDB) 2.81.aq_gt $6$ (not in LMFDB)