Root numbers of a family of elliptic curves and two applications

Abstract

For each $t\in Q\setminus \{-1,0,1\}$, define an elliptic curve over $Q$ by $E_t:y^2=x(x+1)(x+t^2).$Using a formula for the root number $W\left(E_t\right)$ 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 $Z\times Z/2Z\times Z/4Z$, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.