Define the surface $X$ to be the total space of $\mathcal{O}_{\mathbb{P}^1}(-5)$. By contracting the exceptional curve in $X$, we get a surface with an isolated singularity. I am looking for the equation (or the set of equations) that describes this singularity (as a surface in some $\mathbb{C}^n$, possibly just $\mathbb{C}^3$).

For example in the case of $X$ being the total space of $\mathcal{O}_{\mathbb{P}^1}(-2)$, the resulting singularity is the $A_1$ singularity given by $$x^2+yz=0$$.