Properties

 Label 2.81.abd_nx 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: $1 - 29 x + 361 x^{2} - 2349 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.0405552772615$, $\pm0.287450300601$ Angle rank: $2$ (numerical) Number field: 4.0.26125.1 Galois group: $D_{4}$ Jacobians: 20

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 20 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4545 42273045 282413396745 1853030515239045 12157415098767450000 79766031668791568059845 523347276925733859867609945 3433683642766367124221468155845 22528399528490497335275231503415745 147808829467835157825336903404659200000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 53 6443 531413 43046963 3486712598 282428079803 22876776888893 1853020093048163 150094635187410413 12157665463456559198

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.26125.1.
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_nx $2$ (not in LMFDB)