Two-Torsion of Elliptic Curves and Their Twists

Question

Let $E/\mathbb{Q}$ be an elliptic curve such that $E(\mathbb{Q})[2]=\{\mathcal{O}\}$. Does there exist a quadratic twist $E^{(d)}$ of $E$ with a $\mathbb{Q}$-rational two torsion point? I'm particularly interested in the case where $E$ has complex multiplication.

Answer 1

If $E : y^2 = x^3 + ax + b$, then the two torsion points are exactly of the form $(r, 0)$ where $r$ is a root of $x^3 + ax + b$. So your question becomes, if $x^3 + ax + b$ has no rational roots, is there an integer $d$ so that $x^3 + ad^2 x + bd^3$ has rational roots? But the roots of the right hand side are just $dr$, and so it also has no $\mathbb{Q}$-rational two torsion.