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

TL;DR

This paper calculates the rank of specific elliptic curves defined by y^2=x(x-p)(x-2) where p and p-2 are twin primes, using 2-descent, contributing to the classification of elliptic curves by rank.

Contribution

It applies 2-descent to determine the rank of elliptic curves with twin prime parameters, a novel approach for this family.

Findings

Determined the rank for the family of curves with twin prime parameters

Identified conditions under which the rank is positive or zero

Provided explicit rank calculations for specific prime pairs

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.