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