Classification the elements of the twisted Hessian curves in the ring $\mathbb{F}_{q}[e], e^{3}=e^2$

Abstract

Let $\mathbb{F}_{q}$ denote the finite field of $q$ elements, where $q$ is a prime power. In this paper, we study the twisted Hessian curves denoted $H_{a,d}(\mathbb{F}_{q}[e])$ over the ring $\mathbb{F}_{q}[e]$, where $e^{3}=e^2$ and $(a,d)\in (\mathbb{F}_{q}[e])^{2}$. More precisely, we study some arithmetical properties of this ring and using the Twisted Hessian equation, we define the twisted Hessian curves $H_{a,d}(\mathbb{F}_{q}[e])$. This work study the twisted Hessian curve helped us to define two twisted Hessian over the finite field $\mathbb{F}_{q}$. We end this paper by giving the classification of the elements in $H_{a,d}(\mathbb{F}_{q}[e])$.