Ars Combinatoria, cilt.99, ss.519-529, 2011 (SCI-Expanded)
Let p be a prime number and let Fp 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 Ep,λ : y2 = x(x - 1)(x - λ) over Fp for primes p ≡ 3 (mod 4), where λ ≠ 0, 1. We proved that the order of Ep,λ over Fp is p + 1 if λ = 2, p+1/2 or p - 1. Later we generalize this result to F pn 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 Ep,λ over Fp. In the third section, we consider the rank of Eλ: y2 = x(x - 1)(x - λ) over Q.