Arithmetic Relation Between Family of Elliptic Curves Over Finite Field

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Let \(\mathbb{F}_{q}\) be a finite field, where \(q\) is an odd prime such that \(q>3\). Let \(f\left(t\right) =t^{3}-t\) \(\in \mathbb{F}_{q}\left[ t\right]\) be a polynomial of degree 3. For \(\lambda \neq 0\) in \(\mathbb{F}_{q}\), consider families of elliptic curves \(\left\{ E_{\lambda }\right\} _{\lambda \in \mathbb{F}_{q}^{\ast}}\) and \(\left\{ H_{\lambda }\right\} _{\lambda \in \mathbb{F}_{q}^{\ast }}\) defined respectively by\[v^{2}=\lambda f(u)\text{ and }f\left( v\right) =\lambda f(u).\]In this paper, I investigate the relation between the rational points over finite field on \(\left\{E_{\lambda }\left( \mathbb{F}_{q}\right) \right\}_{\lambda \in \mathbb{F}_{q}^{\ast }}\) and \(\left\{ H_{\lambda }\left(\mathbb{F}_{q}\right) \right\} _{_{\lambda \in \mathbb{F}_{q}^{\ast }}}\), and determine the number of rational points on both of these family of curves.