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 $p\equiv 3(\mathrm{mod}4)$, where $\lambda \ne 0,1$. 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 $n\ge 2$. 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 $\mathbb{Q}$.