# Properties

 Label 2.64.aba_ld Base Field $\F_{2^{6}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{6}}$ Dimension: $2$ L-polynomial: $1 - 26 x + 289 x^{2} - 1664 x^{3} + 4096 x^{4}$ Frobenius angles: $\pm0.0466571603306$, $\pm0.280701937272$ Angle rank: $2$ (numerical) Number field: 4.0.1088.2 Galois group: $D_{4}$ Jacobians: 21

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 21 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2696 16380896 68712350792 281487809821568 1152890210522185416 4722323806521178206944 19342785219848857670346248 79228152082414704039889874432 324518552853931502099526217320968 1329227997560012002030049950440159456

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 39 3999 262119 16777983 1073712679 68718855711 4398040168743 281474939649279 18014398464823527 1152921506146497439

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The endomorphism algebra of this simple isogeny class is 4.0.1088.2.
All geometric endomorphisms are defined over $\F_{2^{6}}$.

## 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 2.64.ba_ld $2$ (not in LMFDB)