Let $\mathbb{F}_q$ be a finite field with $q$ elements. For $a,b,c,d,e,f \in \mathbb{F}_q^{\times}$, denote by $C_{a,b,c,d,e,f}$ the family of algebraic curves over $\mathbb{F}_q$ given by the affine equation \begin{align*} C_{a,b,c,d,e,f}:ay^2+bx^2+cxy=d+ex^2y^2+fx^3y. \end{align*} The family of generalized twisted Edwards curves is a subfamily of $C_{a,b,c,d,e,f}$. Let $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ denote the number of points on $C_{a,b,c,d,e,f}$ over $\mathbb{F}_q$. In this article, we find certain expressi…