# Properties

 Label 2.81.abe_ow 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: $( 1 - 16 x + 81 x^{2} )( 1 - 14 x + 81 x^{2} )$ Frobenius angles: $\pm0.151478024726$, $\pm0.216346895939$ Angle rank: $2$ (numerical) Jacobians: 32

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 32 curves, and hence is principally polarizable:

• $y^2=(2a^3+a+1)x^6+(2a^3+a^2+a+1)x^5+(a^3+2a^2+a)x^4+(2a+2)x^3+(2a^2+1)x^2+(2a^2+a+1)x+a^3+a+1$
• $y^2=(2a^3+a)x^6+(a^3+2)x^5+(a^2+2a+2)x^4+(a+2)x^3+(a^2+2a+2)x^2+(a^3+2)x+2a^3+a$
• $y^2=(a^3+2a+2)x^6+(2a^3+2a)x^5+(a^3+a^2+2a)x^4+2a^2x^3+(2a^3+a^2+2a+2)x^2+(2a^3+a^2+2)x+2a^3+2a^2+a+1$
• $y^2=(2a^3+1)x^6+2a^3x^5+(a^3+2a+2)x^4+(2a^2+a)x^3+(a^3+2a+2)x^2+2a^3x+2a^3+1$
• $y^2=(2a^3+2a^2+a+2)x^6+a^3x^5+a^2x^4+(a^3+2a^2+a+1)x^3+a^2x^2+a^3x+2a^3+2a^2+a+2$
• $y^2=(a^2+2)x^6+(a+2)x^5+(2a^3+2a+1)x^4+(a^3+2)x^3+(2a^3+2a+1)x^2+(a+2)x+a^2+2$
• $y^2=(2a^3+a+1)x^6+(a^3+2a^2+a+2)x^5+(a^2+2)x^4+(2a^3+2)x^3+(2a^3+a^2+2a)x^2+(2a^3+a^2)x+2a+1$
• $y^2=(a^3+2a^2+a+2)x^6+(a^2+a+2)x^5+(2a^3+a)x^4+(a^3+2a^2+a+1)x^3+(2a^3+a)x^2+(a^2+a+2)x+a^3+2a^2+a+2$
• $y^2=(2a^3+2)x^6+(a+1)x^5+(2a^3+a)x^4+(a^3+a^2)x^3+(2a^3+a)x^2+(a+1)x+2a^3+2$
• $y^2=(a^3+a^2+a+2)x^6+(a^3+a^2+a)x^5+(2a^3+a^2+2a+2)x^4+(a^3+2a^2+a+1)x^3+(a+2)x^2+a^3x+2a^3+a$
• $y^2=(a^2+2)x^6+(a^3+a^2+a+1)x^5+(a^2+2a)x^4+2a^2x^3+(a^2+2a)x^2+(a^3+a^2+a+1)x+a^2+2$
• $y^2=(2a^3+a^2+a)x^6+(2a^3+2)x^5+(a^3+2a^2+a)x^4+2a^2x^3+(2a^3+2a+1)x^2+(2a^3+a+2)x+a^3+a^2+a+1$
• $y^2=(a^3+a)x^6+2a^3x^5+(2a^3+2a^2+2a)x^4+(2a^3+a^2+2a+2)x^3+(a^3+2a^2+2a+2)x^2+(2a^3+a^2+2)x+2a^3+a^2+2a+1$
• $y^2=(a^3+2a+2)x^6+a^3x^5+(a^3+2a^2+a+1)x^4+(a^3+a^2+2)x^3+(2a^3+2a^2+a)x^2+(2a^3+2a^2+2a)x+a^2+a$
• $y^2=(2a^3+2a+1)x^6+(2a^3+2a+2)x^5+(2a^3+a^2+a+1)x^4+(a^2+a)x^3+(2a^3+a^2+a+1)x^2+(2a^3+2a+2)x+2a^3+2a+1$
• $y^2=(2a^3+a+1)x^6+(2a^3+2a)x^5+a^3x^4+(2a^3+a^2)x^3+a^3x^2+(2a^3+2a)x+2a^3+a+1$
• $y^2=(a^3+2a^2+2a+2)x^6+(2a^3+a^2+2a+1)x^5+(2a^3+a^2)x^4+(2a^2+1)x^3+2a^3x^2+(2a^2+a+1)x+a^3+a$
• $y^2=(a^3+2a)x^6+a^2x^5+(2a^3+a^2+a)x^4+2x^3+(2a^3+a^2+a)x^2+a^2x+a^3+2a$
• $y^2=(2a^3+2a+2)x^6+(a^3+2a^2)x^5+(2a+1)x^4+(a^3+2a^2+a+1)x^3+(2a+1)x^2+(a^3+2a^2)x+2a^3+2a+2$
• $y^2=(a^3+1)x^6+(2a^3+a^2+2)x^5+(a^3+a^2)x^4+(2a^3+2a+1)x^3+(a^3+a^2)x^2+(2a^3+a^2+2)x+a^3+1$
• and 12 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4488 42223104 282669611400 1853719921557504 12158361612293998728 79766908955767535040000 523347838028398318931175048 3433683839993696966739140542464 22528399478664295376291937646110600 147808829336564811862189637126571677184

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 52 6434 531892 43062974 3486984052 282431186018 22876801416052 1853020199483774 150094634855445172 12157665452659227554

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.aq $\times$ 1.81.ao 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.ac_ack $2$ (not in LMFDB) 2.81.c_ack $2$ (not in LMFDB) 2.81.be_ow $2$ (not in LMFDB)