Generators and splitting fields of certain elliptic K3 surfaces

Let $k\subset \mathbb{C}$ be a number field and $\mathcal{E}$ be an elliptic curve defined over $k(t)$, the rational function field of the projective line ${\mathbb{P}}_k^1$, is isomorphic to the generic fiber of an elliptic surface $\pi :={\text{Sc}}_{\text{Ee}}\to {\mathbb{P}}_k^1$. For any subfield $\mathcal{K}\subseteq \mathbb{C}$ of $k$, the set $\mathcal{E}(\mathcal{K}(t))$ of $\mathcal{K}(t)$-rational points of $\mathcal{E}$ is known to be a finitely generated abelian group. The splitting field of $\mathcal{E}$ defined over $k(t)$ is the smallest finite extension $\mathcal{K}\subset \mathbb{C}$ of $k$ such that $\mathcal{E}(\mathbb{C}(t))\text{iso}\mathcal{E}(\mathcal{K}(t))$. In this paper, we consider the elliptic $K3$ surfaces defined over $k=\mathbb{Q}$ with the generic fiber given by the Weierstrass equation ${\mathcal{E}}_n:{\displaystyle y^2=x^3+t^n+1\mathrm{/}{t}^n}$, $1\le n\le 6$, and determine the splitting field ${\mathcal{K}}_n$, and find an explicit set of independent generators for ${\mathcal{E}}_n({\mathcal{K}}_n(t))$ for $1\le n\le 6$.