An elliptic Calabi-Yau threefold of Mordell-Weil rank twelve

We construct a smooth projective Calabi–Yau threefold with a flat elliptic fibration of Mordell–Weil rank twelve and Hodge numbers $(h^{1,1},h^{2,1})=(25,1)$. The construction starts from the Kummer surfaces associated with the products of a fixed elliptic curve and the level-four modular family. The threefold is birational to an anticanonical quartic in $\mathbb{P}({\mathcal{O}}_{{\mathbb{P}}^1}(3{)}^{\oplus 2}\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(4)\oplus {\mathcal{O}}_{{\mathbb{P}}^1})$. We give twelve independent sections and construct a projective crepant resolution of the associated Weierstrass model.