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.