  Let $f: W \rightarrow T$ be an elliptic threefold that is a Weierstrass
model, which is locally defined by $y^2 = x^3 + fx + g$ over $T$, with a
singular fiber such that $(f,g,4f^3 + 27g^2)$ vanishes of order $(4,6,12)$ over
an isolated point over $T$. Such a fiber can be explicitly resolved to fibers
with smaller vanishing order and the resulting model, $Y$, contains a rational
elliptic surface, $S$, where some sections of $S$ are flopping curves on $Y$.
As a consequence of this arithmetic and geometric connection, we are able to
describe some constraints between flops of $Y$ and the properties of the
Mordell-Weil group of $S$ and $Y$.
