27 lines on Fermat cubic

Question

Fermat cubic is $S=\{(x:y:z:w)|x^3+y^3+z^3+w^3 =0\} \in \mathbb{P}^3$. It is obvious that 27 lines on Fermat cubic are represented by $(x,ax,z,bz)$ for cube root $a,b$ of $-1$ and their conjugates. But is there any way to find them by resultant? I tried to follow Hulek's book but it starts by defining $P \in S$ such that $T_P S \cap S=C_P$ has a cusp. Unfortunately, $(1:-1:0:0)$ and some trivial points on $S$ seem to not satisfy that.