The splitting field and generators of the elliptic surface $Y^2=X^3 +t^{360} +1$

TL;DR

This paper introduces a symbolic algorithm to determine the splitting field and generators of the Mordell-Weil lattice for a specific elliptic surface with the highest known rank, combining decomposition techniques and explicit polynomial computations.

Contribution

It provides a novel algorithmic approach to compute the splitting field and generators for a high-rank elliptic surface, expanding understanding of its arithmetic structure.

Findings

Computed defining polynomials of the splitting field with degrees 1728 and 5760.

Identified 68 linearly independent generators for the Mordell-Weil lattice.

Verified results using height pairing matrices and symbolic software.

Abstract

The splitting field of an elliptic surface $\mathcal{E}/\mathbb{Q}(t)$ is the smallest finite extension $\mathcal{K} \subset \mathbb{C}$ such that all $\mathbb{C}(t)$-rational points are defined over $\mathcal{K}(t)$. In this paper, we provide a symbolic algorithmic approach to determine the splitting field and a set of $68$ linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface $Y^2=X^3 +t^{360} +1$. This surface is noted for having the largest known rank 68 for an elliptic curve over $\mathbb{C}(t)$. Our methodology utilizes the known decomposition of the Mordell-Weil Lattice of this surface into Lattices of ten rational elliptic surfaces and one $K3$ surface. We explicitly compute the defining polynomials of the splitting field, which reach degrees of 1728 and 5760, and verify the results via height pairing matrices and specialized symbolic software packages.