#### Abstract

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.

#### Sponsor Name

Liljana Babinkostova

#### Recommended Citation

Kuehnert, Ben; Schlafly, Geneva; and Yi, Zecheng
(2022)
"On Isomorphic K-Rational Groups of Isogenous Elliptic Curves Over Finite Fields,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 23:
Iss.
1, Article 4.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/4

#### Included in

Algebra Commons, Algebraic Geometry Commons, Number Theory Commons