How can I check that a curve inside of $\Bbb P^3$ is an elliptic curve? Specifically, let $C$ be the plane cubic $$C:aX^3+bY^3+cZ^3=0$$ and $\phi:\Bbb P^2\to \Bbb P^3$ given by $[X,Y,Z]\mapsto[X^3,Y^3,Z^3,XYZ]$. How do I put $\phi(C)$ in Weierstrass form and find the degree of $\phi|_C$?
Attempt:
Writing down the induced morphism $\phi^{\#}$ of the function fields $$\begin{aligned}\phi^{\#}:k[x,y,z,w]&\to k[X,Y,Z] \\x&\mapsto X^3\\y&\mapsto Y^3\\z&\mapsto Z^3\\w&\mapsto XYZ\end{aligned}$$we see that $\phi(C)\subset V(ax+by+cz)$. So it is inside a hyperplane, and it suffices to put $C$ in Weierstrass form there. Now, how do I find more equations cutting out $\phi(C)$, and why does it not suffice to find one that pulls back to the generator of $I(V(C))$, which is $C$?