The splitting fields and Generators of Shioda's elliptic surfaces $y^2=x^3+t^m+1$ (I)

The splitting field of an elliptic surface $\mathcal{E}$ defined over $\mathbb{Q}(t)$ is the smallest subfield $\mathcal{K}$ of $\mathbb{C}$ such that $\mathcal{E}(\mathbb{C}(t))\cong \mathcal{E}(\mathcal{K}(t))$. In this paper, we determine the splitting field ${\mathcal{K}}_m$ and a set of linearly independent generators for the Mordell–Weil lattice of Shioda’s elliptic surface with generic fiber given by ${\mathcal{E}}_m:y^2=x^3+t^m+1$ over $\mathbb{Q}(t)$ for positive integers $1\le m\le 12$.