Classify the singular projective surface $y^4 - 4x^3w + 8w^4 - z^2w^2 = 0$

I am seeking guidance on how to properly classify the singular projective surface $S \subset \mathbb{P}^3$ defined by the degree 4 homogeneous polynomial $y^4 - 4x^3w + 8w^4 - z^2w^2 = 0$, which is known to contain some elliptic curve of rank 1, alongside a unique isolated singularity at $P_0 = [0 : 0 : 1 : 0]$.

\begin{aligned} x(u,v) &= -uv^{2}(u^{3}+18v^{3}) \\ y(u,v) &= (u^{3}+21v^{3})(u^{3}+18v^{3}) \\ z(u,v) &= 3u^{6}+110u^{3}v^{3}+1017v^{6} \\ w(u,v) &= (u^{3}+18v^{3})^{2} \end{aligned}

Specifically, I would like to clarify whether a full resolution of singularities $\pi: \tilde{S} \to S$ is strictly required before evaluating classical birational invariants such as the Geometric Genus ($p_g$), Irregularity ($q = h^{0,1}$), Plurigenera ($P_m$), and Kodaira Dimension ($\kappa$), given that the Enriques–Kodaira classification is formulated for smooth minimal models and uneliminated singularities distort canonical line bundles and differential forms. Furthermore, I wish to confirm if Castelnuovo’s Rationality Criterion ($q = 0, P_2 = 0$) definitively rules out $S$ being a rational surface due to the existence of a non-zero global 2-form yielded by the adjoint linear system $\left\vert{} K_{\mathbb{P}^3} + S \right\vert{} = \left\vert{} 0 \cdot H \right\vert{}$, which gives $p_g(\tilde{S}) = 1$ and $P_2(\tilde{S}) \ge 1$, thereby categorizing $\tilde{S}$ as a K3 surface ($\kappa = 0, p_g = 1, q = 0$). Finally, I would appreciate a precise description of the algorithmic workflow in computer algebra systems such as Magma or SageMath/Singular to construct the blow-up tree at $P_0$, execute the resolution of singularities, and compute the invariants of the smooth model $\tilde{S}$.