I want to calculate the singular points of the affine curve $$f(X,Y)=(1+X^2)^2-XY^2 \in \mathbb{C}[X,Y]$$
The point $P=(x,y)$ is singular $\Leftrightarrow$
If $x=0$ we find $y=0$ and then from the first equation we get that $0=1$, that is a contradiction.
For $y=0$, from the second equation we get $x=0$, that we reject, or $x=\pm i$.
The first equation is satified for $y=0$ and $x=\pm i$.
So does this mean that we have the following two singular points:
$$P_1=(i,0) \text{ and } P_2=(-i,0)$$
? Or have I done something wrong?