# Properties

 Label 2.4.ab_d 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: $1 - x + 3 x^{2} - 4 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.254152667512$, $\pm0.647800160692$ Angle rank: $2$ (numerical) Number field: 4.0.46305.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 15 375 3780 76875 1143825 16254000 265693695 4289701875 68323148460 1099644759375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 22 61 298 1114 3967 16216 65458 260629 1048702

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.46305.1.
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_d $2$ 2.16.f_bh