Does 0 lie on elliptic curve?

Question

Does $0$ lie on an elliptic curve, where $0$ is the identity (e.g. $p + 0 = p$)?

Answer 1

Every elliptic curve $E$ over a field $k$ is isomorphic to the locus in $\mathbf{P}_k^2$ cut out by a nonsingular Weierstrass equation in such a way that the given rational point on $E$ is mapped the point at infinity. So I guess the answer to your question is yes.