I'm not sure whether this is obvious or not. The curve $X(1)$ parametrices all elliptic curves up to isomorphism class over $\mathbb{ C}$, and a $K$-rational point corresponds to an elliptic curve whose $j$-invariant lies in $K$, but if I remember correctly the $K$-rational points actually classify elliptic curves up to isomorphism over $K$. Is there an obvious reason for this?