# Properties

 Label 2.81.abd_ny 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: $1 - 29 x + 362 x^{2} - 2349 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.0580440523198$, $\pm0.284000150427$ Angle rank: $2$ (numerical) Number field: 4.0.3516652.1 Galois group: $D_{4}$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4546 42286892 282459800200 1853113095980800 12157515710003257106 79766125852891924985600 523347350858005750641704786 3433683697717156530003068083200 22528399573470900792358054980545800 147808829509839558975231622091913144812

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 53 6445 531500 43048881 3486741453 282428413282 22876780120653 1853020122702881 150094635487090700 12157665466911531805

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.3516652.1.
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_ny $2$ (not in LMFDB)