Mordell-Weil lattice of Inose's elliptic K3 surface arising from the product of 3-isogenous elliptic curves

From the product of two elliptic curves, Shioda and Inose [6] constructed an elliptic K3 surface having two II⁎ fibers. Its Mordell-Weil lattice structure depends on the morphisms between the two elliptic curves. In this paper, we give a method of writing down generators of the Mordell-Weil lattice of such elliptic surfaces when two elliptic curves are 3-isogenous. In particular, we obtain a basis of the Mordell-Weil lattice for the singular K3 surfaces X[3,3,3], X[3,2,3] and X[3,0,3].