As my final project for Brown's MATH1530: Abstract Algebra I investigated elliptic curves. These are some of the most fascinating mathematical objects, as algebraic properties can be applied to the set of points defining the curve. I explore and prove the group structure that can be placed on these curves, as well as explore their critical use in cryptographic systems. This repository houses the necessary .tex, .cls, and media files needed to compile the document, as well as some SageMath code to draw and analyze the curves.