Root numbers of a family of elliptic curves and two applications

For each t∈Q∖{−1,0,1}, define an elliptic curve over Q by Et:y2=x(x+1)(x+t2).Using a formula for the root number W(Et) 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/Q whose Mordell-Weil group contains Z×Z/2Z×Z/4Z, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.