# Properties

 Label 2.81.abd_og Base Field $\F_{3^{4}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

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## Invariants

 Base field: $\F_{3^{4}}$ Dimension: $2$ L-polynomial: $( 1 - 16 x + 81 x^{2} )( 1 - 13 x + 81 x^{2} )$ Frobenius angles: $\pm0.151478024726$, $\pm0.243120792737$ Angle rank: $2$ (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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4554 42397740 282831106536 1853767579023360 12158284207270731354 79766769864367266912000 523347718290325243344640314 3433683788631360922475376599040 22528399496782612690260060577140456 147808829392230024726910141358664283500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 53 6461 532196 43064081 3486961853 282430693538 22876796182013 1853020171765601 150094634976157796 12157665457237837901

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.aq $\times$ 1.81.an 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.ad_abu $2$ (not in LMFDB) 2.81.d_abu $2$ (not in LMFDB) 2.81.bd_og $2$ (not in LMFDB)