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Let $S$ be an Enriques surface over complex numbers. It is known that $S$ admits an elliptic fibration over $\mathbb{P}^1$ with $12$ nodal singular fibers and $2$ double fibers. How can I see this well-known fact? Is it possible to explicitly construct such a fibration via the Enriques lattice $NS(S)\cong U\oplus E_8$?

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A complete and detailed treatment of elliptic pencils on Enriques surfaces (included the result you are quoting) can be found in the book by Barth-Hulek-Peters-Van de Ven Compact Complex Surfaces (Ergebnisse der Mathematik und ihrer Grenzgebiete 4, Springer).

See in particular Chapter VIII, Section 17 "Elliptic pencils".

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