# Properties

 Label 2.81.abc_ng 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: $1 - 28 x + 344 x^{2} - 2268 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.0539942730482$, $\pm0.306978860048$ Angle rank: $2$ (numerical) Number field: 4.0.6334720.3 Galois group: $D_{4}$ Jacobians: 16

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 16 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4610 42421220 282503201810 1853006571051280 12157348072029135250 79766021494260220470500 523347355822296617516069890 3433683764381938628509465067520 22528399625284992067353363690641090 147808829507709580004996979704132310500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 54 6466 531582 43046406 3486693374 282428043778 22876780337654 1853020158679166 150094635832300182 12157665466736335426

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.6334720.3.
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.bc_ng $2$ (not in LMFDB)