Degree Project in Engineering Physics, First Level Department of Mathematics KTH Royal Institute of Technology

Sammanfattning: 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.