The rank and generators of Kihara's elliptic curve with torsion $\mathbb{Z}/4\mathbb{Z}$ over $\mathbb{Q}(t)$

TL;DR

This paper precisely determines the rank and generators of Kihara's elliptic curve over $Q(t)$ with torsion group $Z/4Z$, establishing the current record rank for such curves through specialization techniques.

Contribution

It provides the exact rank and generators of a high-rank elliptic curve over $Q(t)$, improving understanding of elliptic curves with specific torsion structures.

Findings

Exact rank of the curve is determined.

Generators of the Mordell-Weil group are explicitly found.

The curve's rank exceeds previous known records for similar torsion groups.

Abstract

For the elliptic curve $E$ over $\mathbb{Q}(t)$ found by Kihara, with torsion group $\mathbb{Z}/4\mathbb{Z}$ and rank $\geq 5$, which is the current record for the rank of such curves, by using a suitable injective specialization, we determine exactly the rank and generators of $E(\mathbb{Q}(t))$.