  For each $t\in\mathbb{Q}\setminus\{-1,0,1\}$, define an elliptic curve over
$\mathbb{Q}$ by \begin{align*}
  E_t:y^2=x(x+1)(x+t^2). \end{align*} Using a formula for the root number
$W(E_t)$ as a function of $t$ and assuming some standard conjectures about
ranks of elliptic curves, we determine (up to a set of density zero) the set of
isomorphism classes of elliptic curves $E/\mathbb{Q}$ whose Mordell-Weil group
contains $\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times
\mathbb{Z}/4\mathbb{Z}$, and the set of rational numbers that can be written as
a product of the slopes of two rational right triangles.
