  For a given group $G$ and an elliptic curve $E$ defined over a number field
$K$, I discuss the problem of finding $G$-extensions of $K$ over which $E$
gains rank. I prove the following theorem, extending a result of Fearnley,
Kisilevsky, and Kuwata: Let $n = 3,4,$ or $6$. If $K$ contains its
$n^{th}$-roots of unity then, for any elliptic curve $E$ over $K$, there are
infinitely many $\mathbb{Z}/n\mathbb{Z}$-extensions of $K$ over which $E$ gains
rank.
