#### Date of Submission

Spring 2011

#### Academic Program

Mathematics

#### Advisor

John Cullinan

#### Abstract/Artist's Statement

Elliptic curves are cubic curves that have been studied throughout history. From Diophantus of Alexandria to modern-day cryptography, Elliptic Curves have been a central focus of mathematics. This project explores certain geometric properties of elliptic curves defined over finite fields.

Fix a finite field. This project starts by demonstrating that given enough elliptic curves, their union will contain every point in the affine plane. We then find the fewest curves possible such that their union still contains all these points. Using some of the tools discussed in solving this problem, we then explore what can be said about the number of solutions for a particular class of elliptic curves.

#### Distribution Options

Open Access

#### Recommended Citation

McGrath, Travis, "Elliptic Curves: Minimally Spanning Prime Fields and Supersingularity" (2011). *Senior Projects Spring 2011*. 21.

http://digitalcommons.bard.edu/senproj_s2011/21