#### Document Type

Honors Project

#### Abstract

The study of elliptic curves grows out of the study of elliptic functions which dates back to work done by mathematicians such as Weierstrass, Abel, and Jacobi. Elliptic curves continue to play a prominent role in mathematics today. An elliptic curve E is defined by the equation, y^{2} = x^{3} + ax + b, where a and b are coefficients that satisfy the property 4a^{3} + 27b^{2} = 0. The rational solutions of this curve form a group. This group, denoted E(Q), is known as the Mordell-Weil group and was proved by Mordell to be isomorphic to Z^{r} ⊕ E(Q)tors where the group of rational torsion points consists of all points of finite order. The rank r is difficult to compute and the main goal of this research is to explore the relationship between ranks of elliptic curves and values of a and b. Specifically, we have put a lower bound on the ranks of equations of the form C_{m} : y^{2} = x^{3} − m^{2} x + 1 and K_{m} : y^{2} = x^{3} + m^{3} x − m^{3}.

#### Recommended Citation

Bocovich, Cecylia, "Elliptic Curves of High Rank" (2012). *Mathematics, Statistics, and Computer Science Honors Projects.* Paper 24.

http://digitalcommons.macalester.edu/mathcs_honors/24

© *Copyright is owned by author of this document*

## Comments

Thank you to my advisor, Dr. Bressoud, and to my readers, Dr. Doyle and Dr. Flath.