Let $Y$ be the smooth manifold underlying a K3 surface. As a manifold, $Y$ is diffeomorphic to $\{[x_0:x_1:x_2:x_3]\in\mathbb{C}P^3\colon X_0^4+x_1^4+X_2^4+X_3^4=1\}$. It is well known that $H^2(Y,\mathbb{Z})\cong U^{\oplus 3}\oplus(-E_8)^{\oplus 2}\cong \mathbb{Z}^{\oplus 22}$.

From the classification results of principal $S^1$-bundles $p:M\to Y$, we have know that they are classified by $H^2(Y,\mathbb{Z})$. Namely, by picking a connection 1-form $\theta$ on $M$, which satisfies $d\theta=p^*\omega$ for some closed $\omega\in \Omega^2(Y)$, we get a map $$\Phi:\{\text{Isomorphism classes of principal $S^1$-bundles $p:M\to Y$}\}\to H^2(Y,\mathbb{Z}),$$ by sending $[p:M\to Y]$ to $[\omega]$.

In the case that $Y$ is a K3 surface, I am wondering if there are concrete presentations of these principal $S^1$-bundles around. More concretely, suppose we have have a basis $c_1,\cdots,c_{22}$ of $H^2(Y,\mathbb{Z})$, can we construct 22 principal $S^1$-bundles with classes $c_1,\cdots,c_{22}$.

Since $S^1\to S^7\to \mathbb{C}P^3$ is an $S^1$-bundle, I tried to realize $Y$ as an $S^1$-bundle $S^1\to M\to Y$, where $M\subset S^7$ is some smooth submanifold. But this is probably not going to give me all bundles over $Y$, so any help or any link to literature would be helpful!