Characterizing terminal Fano threefolds with the smallest anti-canonical volume

It was proved by J. A. Chen and M. Chen that a terminal Fano $3$-fold $X$ satisfies $(-K_X)^3geq frac{1}{330}$. We show that a non-rational $mathbb{Q}$-factorial terminal Fano $3$-fold $X$ with $rho(X)=1$ and $(-K_X)^3=frac{1}{330}$ is a weighted hypersurface of degree $66$ in $mathbb{P}(1,5,6,22,33)$.