# Properties

 Label 2.16.ak_bx Base Field $\F_{2^{4}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $2$ L-polynomial: $1 - 10 x + 49 x^{2} - 160 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.0660425289118$, $\pm0.412497962872$ Angle rank: $2$ (numerical) Number field: 4.0.10304.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 136 64736 16734664 4257298304 1096317656776 281420217561056 72064807316490376 18447125055386607104 4722354706500636490504 1208924802784313506001376

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 255 4087 64959 1045527 16773951 268462327 4295055999 68719305367 1099510702975

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.10304.1.
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## 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 2.16.k_bx $2$ 2.256.ac_alb