I am reading this paper: https://arxiv.org/abs/0810.2687 by A. J. de Jong and Robert Friedman. In the proof of Theorem 4.10, a singularity of the following type shows up $$y^2=x^3+z^{6d-1}.$$ When $d=1$, this is exactly the type of $E_8$ in Du Val singularities. And the Milnor lattice and Dynkin diagram can be obtained by blowing up. In general, looks like the corresponding Milnor lattice is isomorphic to $(2d-2)U\oplus d(-E_8)$.
I am wondering how to verify this in this case. In particular, can this be deduced from the case when $d=1$? Thanks!