  This paper is concerned with the construction of extremal elliptic K3
surfaces. It gives a complete treatment of those fibrations which can be
derived from rational elliptic surfaces by easy manipulations of their
Weierstrass equations. In particular, this approach enables us to find explicit
equations for 38 semi-stable extremal elliptic K3 fibrations, 32 of which are
indeed defined over the rationals. They are realized as pull-back of
non-semi-stable extremal rational elliptic surfaces via base change. This is
related to work of J. Top and N. Yui which exhibited the same procedure for the
semi-stable extremal rational elliptic surfaces.
