Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular fiber of Kodaira Type $I_{1}$ and $II$, and $m+2n=24$ where $n (m)$ is the number of singular fiber of Kodaira Type $I_{1}(II)$.

Can we say something similar when $\mathrm{Pic}(S)\cong U(k)$ for some $k>1$?