The Elliptic Curves y 2 = x 3 − t 2 x over F p

Let p be a prime number, Fp be a finite field and t ∈ Fp = Fp − {0}. In this paper we obtain some properties of elliptic curves Ep,t : y = y = x − tx over Fp. In the first section we give some notations and preliminaries from elliptic curves. In the second section we consider the rational points (x, y) on Ep,t. We give a formula for the number of rational points on Ep,t over Fp for an integer n ≥ 1. We also give some formulas for the sum of x−and y−coordinates of the points (x, y) on Ep,t. In the third section we consider the rank of Et : y = x − tx and its 2−isogenous curve Et over Q. We proved that the rank of Et and Et is 2 over Q. In the last section we obtain some formulas for the sums ∑ t∈F∗p n p,t for an integer n ≥ 1, where ap,t denote the trace of Frobenius. Keywords—elliptic curves over finite fields, rational points on elliptic curves, rank, trace of Frobenius.