We study the global root number of the complex $L$-function of twists of elliptic curves over $\mathbb{Q}$ by real Artin representations. We obtain examples of elliptic curves over $\mathbb{Q}$ which, while not having any rational points of infinite order, conjecturally must have points of infinite order over the fields $\mathbb{Q}( \sqrt[3] {m} )$ for every cube-free $m > 1$. We describe analogous phenomena for elliptic curves over the fields $\mathbb{Q}( \sqrt[r] {m} )$, and in the towers $(\mathbb{Q}( \sqrt[r^n] {m})_{n \ge 1} )$ and $(\mathbb{Q}( \sqrt[r^n] {m}, \mu_{r^n})_{n \ge 1})$, where $r \ge 3$ is prime.