Is the coordinate ring of an elliptic curve principal?

Question

Let $K$ a field, $E$ an elliptic curve. I would like to know if the coordinate ring of $K[E]=K[X,Y]/(E)$ is principal. I think the answer is no. I tried to prove that the ideal $J=\langle y,x^2\rangle$ which is included in $K[E]$ is not principal but this ideal is $K[E].$ So iam stuck. By advance thank you.