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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 areinfinitely many $\mathbb{Z}/n\mathbb{Z}$-extensions of $K$ over which $E$ gainsrank.

Ranks of elliptic curves over cyclic cubic, quartic, and sextic extensions