# Properties

 Label 2.4.ac_b Base Field $\F_{2^{2}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{2}}$ Dimension: $2$ L-polynomial: $1 - 2 x + x^{2} - 8 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.0935673124239$, $\pm0.651114279890$ Angle rank: $2$ (numerical) Number field: 4.0.1088.2 Galois group: $D_{4}$ Jacobians: 3

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 3 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 8 224 2696 65408 1089288 16380896 271193672 4345445888 68712350792 1101692811744

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 15 39 255 1063 3999 16551 66303 262119 1050655

## Decomposition and endomorphism algebra

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

## 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 2.4.c_b $2$ 2.16.ac_b