I have the triple branched covering $X$ of $\mathbb{P}^{1}$ defined by $y^{3}=x^{6}-1$. I want to show the following:
(i) The canonical embedding $\phi: X \rightarrow \mathbb{P}^{3}$ can be given in affine coordinates by $\phi(x,y)=(x,x^{2}, y)$.
(ii) Find irreducible quadric and cubic hypersurfaces in $\mathbb{P}^{3}$ that contain $\phi(X)$.
I am rather lost on how to approach this. Any help would be appreciated! Thanks very much.