 Solution

The left side is the norm form $N_{K/\mathbb{Q}}(x+y\alpha +z{\alpha }^2)$ for ${\alpha }^3=7$. Reducing a hypothetical equation modulo 7 and then descending through the forced divisibilities of $x,y,z,t$ gives an infinite descent, so only the zero solution exists. If $N(\theta )=3$, clearing denominators minimally and applying the norm equation would produce a forbidden nonzero integer solution; hence no such $\theta$ exists.