For $f(z,w) \in \mathbb{C}[x,y]$ a square-free polynomial, consider an affine threefold $$ X = V(xy + f(z,w)) \subset \mathbb{A}^4. $$ By computing derivatives one sees that the singularities of $X$ are precisely $(0,0,z,w)$ where $(z,w)$ is a singular point of the plane curve $f(z,w) = 0$. Since $f(z,w)$ is assumed to be square-free, singularities of $X$ are isolated.

The question is: what is the reference and/or a computation for the Class group $Cl(X)$?