For $f(z,w) \in \mathbb{C}[z,w]$ 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 $Cl(X)$?

UPDATE: These singularities are typicically non-factorial, that is $Cl(X)$ may be strictly bigger than $Pic(X)$. After seeing Jason Starr's comment, I realized that I am more interested in the quotient $Cl(X) / Pic(X)$ rather than $Cl(X)$ itself.

A particular case of the situation above is when $f(z,w)$ is quasi-homogeneous, so that $\mathbb{C}[X]$ is a graded ring, and $Pic(X) = 0$ but $Cl(X)$ does not have to be zero.