# Properties

 Label 2.81.abd_oa 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: $1 - 29 x + 364 x^{2} - 2349 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.0843104972791$, $\pm0.276447108888$ Angle rank: $2$ (numerical) Number field: 4.0.1141272.2 Galois group: $D_{4}$ Jacobians: 24

This isogeny class is simple and geometrically 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 24 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4548 42314592 282552613632 1853277743837568 12157713899722300548 79766305078966740983808 523347479886540892610608068 3433683777857815302907439973888 22528399625041128513427336385127168 147808829551648849493807957103271412832

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 53 6449 531674 43052705 3486798293 282429047870 22876785760805 1853020165951553 150094635830675450 12157665470350456049

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.1141272.2.
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.bd_oa $2$ (not in LMFDB)