The study of elliptic curves is an important part of the elds of

algebraic geometry and number theory, with many applications in areas

such as cryptography. While much of the groundwork has already

been laid out, the results often times fall short of giving an easily digestible

overview of the subject as a whole. The aim of this paper is

to condense a number of high-level results into a much more readily

accessible version that is better suited for a reader encountering elliptic

curves for the rst time. Additionally, the paper provides a toolkit

for identifying elliptic curve groups, detailing steps to take in order to

determine the group behind a given elliptic curve.