  In previous work, we have shown that elliptic fibrations with two sections,
or Mordell-Weil rank one, can always be mapped birationally to a Weierstrass
model of a certain form, namely, the Jacobian of a $\mathbb{P}^{112}$ model.
Most constructions of elliptically fibered Calabi-Yau manifolds with two
sections have been carried out assuming that the image of this birational map
was a "minimal" Weierstrass model. In this paper, we show that for some
elliptically fibered Calabi-Yau manifolds with Mordell-Weil rank-one, the
Jacobian of the $\mathbb{P}^{112}$ model is not minimal. Said another way,
starting from a Calabi-Yau Weierstrass model, the total space must be blown up
(thereby destroying the "Calabi-Yau" property) in order to embed the model into
$\mathbb{P}^{112}$. In particular, we show that the elliptic fibrations studied
recently by Klevers and Taylor fall into this class of models.
