# Properties

 Label 2.4.ac_h 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: $1 - 2 x + 7 x^{2} - 8 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.293751018564$, $\pm0.533021264456$ Angle rank: $2$ (numerical) Number field: 4.0.10304.1 Galois group: $D_{4}$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 14 476 4802 64736 1089774 16816604 261596594 4257298304 69053293022 1102713976476

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 27 75 255 1063 4107 15963 64959 263415 1051627

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.10304.1.
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.c_h $2$ 2.16.k_bx