$y^2=x^3+f\, x+g$, and the orders of the discriminant, $\Delta$.

When an elliptic fibration has a global section, the set of sections form a group, known as the Mordell–Weil group. The rank of the Mordell–Weil group gives the number of the $U(1)$ gauge fields in F-theory compactification on the elliptic fibration [@MV2].

The second integral cohomology group $H^2(S, \Z)$ of K3 surface $S$ includes the information of the geometry of the K3 surface. This group has the lattice structure, and it is called the K3 lattice, $\Lambda_{K3}$. The K3 lattice is unimodular, even lattice of signature (3,19), and it is isometric to the direct sum of two $E_8$’s and three hyperbolic planes [@Mil] $$\Lambda_{K3} \cong E_8^2 \oplus H^3.$$ The group of divisors (modulo algebraic equivalence) constitutes a sublattice inside the K3 lattice, called the Néron-Severi lattice $NS(S)$. When a K3 surface has an elliptic fibration with a global section, an elliptic fiber and a global section generate the hyperbolic plane $H$ inside the Néron-Severi lattice. K3 surface $S$ admitting an elliptic fibration with a section is equivalent to the condition that the Néron-Severi lattice $NS(S)$ contains the hyperbolic plane $H$ [@Kondoauto]. When an elliptic K3 surface has the singular fibers, the 