The Elliptic Curves \( y^2 = x(x-1)(x-\lambda)\)

Abstract

Let \(p\) be a prime number and let \(\mathbb{F}_p\) 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 \(E_{p,\lambda} : y^2 = x(x-1)(x-\lambda)\) over \(\mathbb{F}_p\) for primes \(p \equiv 3 \pmod{4}\), where \(\lambda \neq 0, 1\). We prove that the order of \(E_{p,\lambda}\) over \(\mathbb{F}_p\) is \(p+1\) if \(\lambda = 2,\frac{p+1}{2}\) or \(p-1\). Later, we generalize this result to \(\mathbb{F}_{p^n}\) for any integer \(n \geq 2\). Also, we obtain some results concerning the sum of \(x\)- and \(y\)-coordinates of all rational points \((x,y)\) on \(E_{p,\lambda}\) over \(\mathbb{F}_p\). In the third section, we consider the rank of \(E_\lambda : y^2 = x(x-1)(x-\lambda)\) over \(\mathbb{Q}\).