Let $X$ be a supersingular K3 surface over an algebraically closed field $k$ of positive characteristic $\!p$. Artin proved in the paper https://eudml.org/doc/81948 that the determinant $\mathrm{disc}(X)$ of a matrix of the intersection pairing on the Néron-Severi group $NS(X)$ is equal to $-p^{2\sigma}$, where $1 \leqslant \sigma \leqslant 10$.

If X has a quasi-elliptic fibration, then $\sigma$ is computable according to http://arxiv.org/pdf/math/0311057.pdf (16 page). What if X has only an elliptic fibration?