I was reading Silverman's Arithmetic of elliptic curves, chapter 3, proposition 3.1, which says that via Riemann Roch one can find functions $x,y$ on an elliptic curve which satisfy a weierstrass equation. One obtains a morphism $\psi : E \rightarrow \mathbb{P}^2$ given by $[x:y:1]$ whose image lies in the zero locus $C$ of this Weierstrass equation. Then Silverman claims this morphism is surjective, the reason being that $C$ is irreducible and the map is non constant.
My question is, why is said locus irreducible?
JyrkiLahtonen's comment answered my question, so im posting it here:
A smooth plane curve is always irreducible (any two components intersect by Bezout, and the points of intersections are automatically singularities). The possible singularities are listed in Proposition 1.4. Proposition 2.5 (among other things) implies that the zero locus is irreducible even in the singular cases. There are other routes to the same conclusion, some probably simpler.