$(*)$ I would like to prove that $Spec(\mathbb{C}[X,Y,Z]/\langle X^2+Y^5+Z^5+1 \rangle)$ is not rational (or equivalently that $Proj (\mathbb{C}[X,Y,Z,T]/\langle X^2 T^3+Y^5+Z^5+T^5 \rangle)$ is not rational).
$(**)$ More generally, I am interested in proving that the hypersurfaces defined by $X^2 + Y^c+Z^c + 1$ where $c \geq 5$ is odd are not rational.
The first thing I tried to do was to consider the projective completion $V(X^2 T^3 + Y^5+Z^5 + T^5) \subset \mathbb{P}^3$. This projective hypersurface has an isolated singular point at $(1:0:0:0)$ and is normal. Magma calculates that this singular projective hypersurface is birational to a smooth projective surface with non-zero geometric genus. This shows (by birational equivalence) that $X^2 + Y^5+Z^5 + 1$ is not rational. The same idea works in the general case. Unfortunately, I have no idea how to calculate the geometric genus by hand, so I haven't been able to prove what I suspect to be true.
Any method of proving $(*)$ or $(**)$ would be useful.
Remark: $Spec(\mathbb{C}[X,Y,Z]/\langle X^2+Y^3+Z^3+1 \rangle)$ is rational.