Elliptic curve terminology confusion

Question

I've been reading a paper that says "Let $E(K)$ be an elliptic curve..." where $E(K)$ means the $K$-rational points of $E$ (where $K$ is a number field). I've seen phrases like "Let $E/K$ be an elliptic curve" and I know that one considers $E(K)$ to be a group and such. But why can I say $E(K)$ is an elliptic curve?