On the Rank of the Elliptic Curve y^2=x(x-p)(x-2)

Abstract

An elliptic curve E defined over \Q is an algebraic variety which forms a finitely generated abelian group, and the structure theorem then implies that E = \Z^r + \Z_{tors} for some r \geq 0; this value r is called the rank of E. It is a classical problem in the study of elliptic curves to classify curves by their rank. In this paper, the author uses the method of 2-descent to calculate the rank of two families of elliptic curves, where E is given by E: y^2 = x(x-p)(x-2) with p, p-2 being twin primes.