# Properties

 Label 2.4.ae_j 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: $1 - 4 x + 9 x^{2} - 16 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.117169895439$, $\pm0.478661301576$ Angle rank: $2$ (numerical) Number field: 4.0.4752.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 6 276 3942 57408 1046166 17589204 273933078 4298940672 68831612838 1103041860756

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 1 19 61 223 1021 4291 16717 65599 262573 1051939

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.4752.1.
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.e_j $2$ 2.16.c_ap