The elliptic curves y2 = x(x - 1)(x - λ)

Abstract

Let p be a prime number and let F-p be a finite field. In the first section, we give some preliminaries from elliptic curves over finite fields. In the second section we consider the rational points on the elliptic curves E-p,E-lambda : y(2) = x(x - 1)(x - lambda) over F-p for primes p equivalent to 3 (mod 4), where lambda not equal 0, 1. We proved that the order of E-p,E-lambda over F-p is p + 1 if lambda = 2, p+1/2 or p - 1. Later we generalize this result to F-p(n) for any integer n >= 2. Also we obtain some results concerning the sum of x-and y-coordinates of all rational points (x, y) on E-p,E-lambda over F-p. In the third section, we consider the rank of E-lambda : y(2) = x(x - 1)(x - lambda) over Q.