I'm trying to compute Neron-Severi lattices of some K3 surfaces. They have elliptic fibrations with multiple sections. Setting one section to be the identity section, I can write down a Weierstrass equation, and compute the singular fibers and Mordell-Weil group. In cases where Mordell-Weil is finite, I can work out NS following the methods outlined in Belcastro's thesis: the fiber and identity section give a copy of $ U = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) $ and the components of the singular fibers that don't intersect the identity section give some root lattices. The quotient of NS by the direct sum of $U$ and the root lattices is the Mordell-Weil group, so we have a finite-index sublattice of NS. Then we apply results of Nikulin about discriminant forms of overlattices, and presto.

My question is what to in the case when the Mordell-Weil group has positive rank. My only idea is to try to write down the intersection matrix explicitly. To do this, I need to know how a section of infinite order intersects the singular fibers. I haven't seen this worked out anywhere. Any ideas?