Ingram, PatrickStevens, Peter Ryan2025-11-112025-11-112025-09-252025-11-11https://hdl.handle.net/10315/43408This thesis investigates the ranks of Ulmer curves over the function fields F_p(t), p a prime, with a focus on computational techniques to estimate their group structure. Using SageMath, we implement point-generation algorithms, discriminant checks, and height-pairing computations to produce numerical evidence supporting predicted ranks. We combine brute-force and probabilistic sampling methods, enabling point generation and verification across a range of parameters. These results illustrate the computational challenges in large rank detection, suggest refinements, and contribute to the broader study of function fields.Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.MathematicsTheoretical mathematicsElectronic Thesis or Dissertation2025-11-11Elliptic CurvesRanksBirch and Swinnerton-Dyer conjectureDouglas UlmerFunction fieldsSageMathMordell-Weil groupElliptic curve cryptography