I am interested in the hypersurface $X\subset\mathbb{A}^4_{\mathbb{F}_{5^n}}$ defined by $$ X = \{x^3 + 3xy^2 + z^3 + 3zw^2 + 1 = 0\} $$ over a finite field $\mathbb{F}_{5^n}$ with $5^n$ elements. Via some computer experiment I have noticed that when $n$ is odd the number of points of $X$ is equal to the number of points of $\mathbb{A}^3_{\mathbb{F}_{5^n}}$.
Is this just a coincidence or is there a theoretical reason for this?
Thank you.
Note that the defining equation for $X$ can be rewritten as $$(x+y)^3+(x-y)^3+(z+w)^3+(z-w)^3=-2.$$
As the linear transformation $(x,y,z,w)\mapsto(x+y,x-y,z+w,z-w)$ is invertible over any field of characteristic not equal to $2$, the number of $\mathbb{F}_{5^n}$-points on $X$ is equal to the number of $\mathbb{F}_{5^n}$-points on the hypersurface $Y$ with defining equation $$a^3+b^3+c^3+d^3=-2.$$
As $3$ does not divide $5^n-1$, every element of $\mathbb{F}_{5^n}$ is a cube in a unique way, so the number of points on $Y$ are the same as the number of points on the hypersurface defined by $$e+f+g+h=-2,$$ but this hypersurface is isomorphic to $\mathbb{A}^3,$ as desired.
