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 $NS(S)\cong U\oplus E\_8$?



A question on an elliptic fibration of the Enriques surface
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". 


