Is it possible to realize K3 $K3$ surface as a ramified double cover of rational elliptic surface? If so, is there way to see an elliptic fibration structure on K3 $K3$ from such cover. ? It seems to me one can use the divisor $6H - 2E_{1} - \cdots - 2E_{9}$ and the class $3H - E_{1} - \cdots - E_{9}$ in two fold cover gives the fiber class in K3? $K3$?
Is it possible to realize K3 surface as a ramified double cover of rational elliptic surface? If so, is there way to see an elliptic fibration structure on K3 from such cover. It seems to me one can use the divisor $6H - 2E_{1} - \cdots - 2E_{9}$ and the class $3H - E_{1} - \cdots - E_{9}$ in two fold cover gives the fiber class in K3?