Splitting Field and Generators of a High-Rank Elliptic Surface

This paper addresses a specific problem in algebraic geometry, focusing on the properties of an elliptic surface with a remarkably high rank (68). The research is significant because it contributes to our understanding of elliptic curves and their associated Mordell-Weil lattices. The determination of the splitting field and generators provides valuable insights into the structure and behavior of the surface. The use of symbolic algorithmic approaches and verification through height pairing matrices and specialized software highlights the computational complexity and rigor of the work.

• •The paper focuses on the elliptic surface defined by $Y^2=X^3 +t^{360} +1$.
• •It determines the splitting field, which is the smallest extension where all rational points are defined.
• •It finds 68 linearly independent generators for the Mordell-Weil lattice, which is a measure of the curve's complexity.
• •The methodology involves decomposing the surface into simpler components and using symbolic computation.
• •The results are verified using height pairing matrices and specialized software.