# Properties

 Label 2.64.ay_kd 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: $( 1 - 15 x + 64 x^{2} )( 1 - 9 x + 64 x^{2} )$ Frobenius angles: $\pm0.113134082257$, $\pm0.309839631512$ Angle rank: $2$ (numerical) Jacobians: 105

This isogeny class is not 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 105 curves, and hence is principally polarizable:

• $y^2+(x^2+x+a^3+a+1)y=(a^5+a)x^5+(a^5+a)x^3+(a^5+a+1)x^2+(a^5+a^4)x+a^4+1$
• $y^2+(x^2+x+a^5+a^3+a^2+a+1)y=(a^4+a^2+a+1)x^5+(a^4+a^2+a+1)x^3+(a^4+a^2+a)x^2+ax+a^5+a^2+a+1$
• $y^2+(x^2+x+a^3+1)y=(a^5+a^4+a^2+1)x^5+(a^5+a^4+a^2+1)x^3+(a^5+a^4+a^2)x^2+(a^5+a^2+1)x+a^4+a^2+1$
• $y^2+(x^2+x)y=(a^5+a^4+a^2+a+1)x^5+(a^4+a^3+a^2)x^3+(a^5+a^3+a^2+1)x^2+(a^2+a)x$
• $y^2+(x^2+x)y=(a^4+a+1)x^5+(a^5+a^4+a^3+a^2)x^3+(a^5+a^4+a^3+a+1)x^2+(a^4+a^2)x$
• $y^2+(x^2+x+a^4+a^3+a^2+1)y=(a^5+a^2+1)x^5+(a^5+a^3+a^2)x^4+(a^5+a^2+a)x^3+(a^5+a^3+a^2)x^2+(a^5+a^2+1)x+a^5+a^3+1$
• $y^2+(x^2+x)y=(a^5+a^4+a)x^5+(a^3+a^2+1)x^3+(a^4+a^3)x^2+(a^5+a^2+a+1)x$
• $y^2+(x^2+x+a^4+a^3+a^2+a+1)y=(a^4+a+1)x^5+(a^5+a^4+a^3+a^2)x^4+(a^5+a^4+a^2)x^3+(a^5+a^4+a^3+a^2)x^2+(a^4+a^3+a)x+a^5+a^2+1$
• $y^2+(x^2+x)y=(a^5+a^4+a+1)x^5+(a^3+a^2+a+1)x^3+(a^5+a^3)x^2+(a^4+a^2)x$
• $y^2+(x^2+x)y=a^4x^5+(a^5+a^3+a)x^4+(a^5+a^4+a^3+a+1)x^3+(a^5+a^4+a^2+1)x^2+(a^5+a^4+a^2)x$
• $y^2+(x^2+x+a^5+a^3+a+1)y=(a+1)x^5+(a^2+1)x^3+(a^5+a^4+a^3+a+1)x+a^4+a^2+a+1$
• $y^2+(x^2+x+a^5+a^4+a^3+a^2+1)y=a^5x^5+(a^5+a^3+a)x^4+(a^5+a^2+1)x^3+(a^5+a^3+a)x^2+a^5x+a^2+a$
• $y^2+(x^2+x)y=(a^5+a)x^5+a^3x^3+(a^2+1)x^2+(a^5+a^3+a^2+a+1)x$
• $y^2+(x^2+x)y=(a^5+a^2+a+1)x^5+(a^3+a+1)x^4+(a^5+a^3+a^2+1)x^3+(a^4+a^3+a^2+a+1)x^2+(a^4+a^3+a^2+a)x$
• $y^2+(x^2+x)y=ax^5+(a^3+a)x^3+(a^5+a^3+a^2)x^2+(a^5+a^2)x$
• $y^2+(x^2+x+a^5+a^4+a^3+a)y=(a^4+a^2+a)x^5+(a^5+a^3+1)x^4+(a^5+a^4)x^3+(a^5+a^3+1)x^2+(a^5+a^4+a^3)x+a^4+a^3+a^2$
• $y^2+(x^2+x+a^5+a^4+a^3+a+1)y=(a^4+a^2+a)x^5+(a^4+a^3+a)x^4+(a^5+a^2+a+1)x^3+(a^4+a^3+a)x^2+(a^3+a^2)x+a^5+a^3+a+1$
• $y^2+(x^2+x+a^5+a^4+a^3+1)y=(a^5+a^4+a)x^5+(a^4+a^3+a^2+1)x^4+(a^4+a^2+a)x^3+(a^4+a^3+a^2+1)x^2+(a^5+a^3)x+a^5+a^4+a^3+a$
• $y^2+(x^2+x)y=(a^5+a^2+a+1)x^5+(a^3+a)x^4+(a^3+a^2+a)x^3+(a^3+1)x^2+(a^5+a^3+a)x$
• $y^2+(x^2+x)y=(a^5+a^4+a^2+a+1)x^5+(a^5+a^3+a^2+a)x^4+(a^4+a^3)x^3+(a^5+a^4+a^3)x^2+(a^5+a^4+a^3+1)x$
• and 85 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2800 16576000 68851627600 281554963200000 1152925088621614000 4722353120372542144000 19342811847298974717936400 79228170911081655733939200000 324518562293698929071779850585200 1329228000411530789994216990366400000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 41 4047 262649 16781983 1073745161 68719282287 4398046223129 281475006542143 18014398988835881 1152921508619795727

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The isogeny class factors as 1.64.ap $\times$ 1.64.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ag_ah $2$ (not in LMFDB) 2.64.g_ah $2$ (not in LMFDB) 2.64.y_kd $2$ (not in LMFDB)