Arithmetic Information of Rational Elliptic Surfaces, and Shioda's Rank 68 Elliptic Surface

The field of definition of the Mordell-Weil group of an elliptic surface $E\to {\mathbb{P}}^1$ defined over $\mathbb{Q}$ is the smallest number field $k$ such that all of its $\text{Bar}\mathbb{Q}(t)$-rational points are defined over $k(t)$. In this paper, we present an algorithm, implemented in \Magma{}, which can determine arithmetic information, including the field of definition, associated to any rational elliptic surface. As an application of this, we also demonstrate that the field of definition of Shioda’s rank $68$ elliptic surface given by $y^2=x^3+t^{360}+1$ is a number field of degree $829,440$.