On the Ranks of Elliptic Curves with Isogenies

In recent years, the question of whether the ranks of elliptic curves defined over Q are unbounded has garnered much attention. One can create refined versions of this question by restricting one’s attention to elliptic curves over Q with a certain algebraic structure, e.g., with a rational point of a given order. In an attempt to gather data about such questions, we look for examples of elliptic curves over Q with an n-isogeny and rank as large as possible. To do this, we use existing techniques due to Rogers, Rubin, Silverberg, and Nagao and develop a new technique (based on an observation made by Mazur) that is more computationally feasible when the naive heights of the elliptic curves are large.