# Properties

 Label 2.64.az_kp 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: $1 - 25 x + 275 x^{2} - 1600 x^{3} + 4096 x^{4}$ Frobenius angles: $\pm0.0763965765331$, $\pm0.298668725704$ Angle rank: $2$ (numerical) Number field: 4.0.3297921.2 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2747 16473759 68771683412 281508684183531 1152897481748456477 4722333178141140465744 19342797421940923903701767 79228162528834195562225281875 324518559038992552084565483210732 1329228000084445085013278452474245279

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 40 4022 262345 16779226 1073719450 68718992087 4398042943180 281474976762418 18014398808163385 1152921508336094102

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The endomorphism algebra of this simple isogeny class is 4.0.3297921.2.
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.z_kp $2$ (not in LMFDB)