I'm not sure whether this is obvious or not. If I remember correctly, 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 $K$, and the $K$-rational points actually classify elliptic curves up to isomorphism over $K$. Is there an obvious reason for this?