# Properties

 Label 2.4.ab_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: $1 - x + x^{2} - 4 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.205814536801$, $\pm0.684666034597$ Angle rank: $2$ (numerical) Number field: 4.0.54665.1 Galois group: $D_{4}$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 13 299 3484 79235 1132573 16667456 272590513 4282572515 68246337484 1099301402979

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 18 55 306 1104 4071 16636 65346 260335 1048378

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.54665.1.
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.b_b $2$ 2.16.b_z