# Properties

 Label 2.81.abf_pk 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: $( 1 - 17 x + 81 x^{2} )( 1 - 14 x + 81 x^{2} )$ Frobenius angles: $\pm0.106600758076$, $\pm0.216346895939$ Angle rank: $2$ (numerical) Jacobians: 14

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 14 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4420 42007680 282364186000 1853405894545920 12158106367827530500 79766748462001176576000 523347775997496339979781380 3433683857278199941704279214080 22528399545595777515915934575706000 147808829423466078703027491480481872000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 51 6401 531318 43055681 3486910851 282430617758 22876798704531 1853020208811521 150094635301373718 12157665459807085601

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.ar $\times$ 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.ad_acy $2$ (not in LMFDB) 2.81.d_acy $2$ (not in LMFDB) 2.81.bf_pk $2$ (not in LMFDB)