This conjecture connects the rank of the group of rational points to the number of points on an elliptic curve mod p and is backed up by a wealth of experimental evidence. Elliptic curves are essential mathematical objects that appear in numerous contexts, including Wiles’ demonstration of the Fermat Conjecture, the factorization of numbers into primes, and cryptography, to name just three. Elliptic curves are described by cubic equations in two variables. Bryan John Birch Professor Sir Peter Swinnerton-Dyer Early History Problems on curves of genus 1 feature prominently in Diophantus’ Arithmetica. It is easy to see that a straight…