It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$-invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of either 0 or 1728, using Vl\u{a}du\c{t}'s characterization of the group structure of rational points.

Author Bio

Ben Kuehnert is studying math and computer science at the University of Rochester. He plans on pursuing a PhD in theoretical computer science.

Geneva Schlafly is a third-year mathematics major at the University of California, Santa Barbara, and beginning her PhD next year. Her main research focuses are algebraic number theory and medical imaging.

Zecheng Yi is a senior at Johns Hopkins University. He is pursuing a B.A./M.A. combined degree in pure mathematics. His major research interest lies in number theory and representation theory, and will continue this focus towards a PhD.