Horizontal tangents to an implicitly defined function

Consider the implicit function: y-y^5=x^4-2x^3+x^2

dy/dx = 2x(2x^2-3x+1)/1-5y^4

Solving for dy/dx=0, there are 3 solutions, namely x=0,1/2 and 1.

This implies that there are 3 horizontal tangents to y-y^5=x^4-2x^3+x^2.

But i suspect that there are more than 3 horizontal tangents to the curve, since a circle given by the equation x^2+y^2=1 has two horizontal tangents at

(0,1) and (0,-1), even though the derivative -x/y has only one solution. Is there any way that this can be shown? Or I am wrong?