Ranks of elliptic curves in cyclic sextic extensions of $\mathbb{Q}$

For an elliptic curve $E\mathrm{/}\mathbb{Q}$ we show that there are infinitely many cyclic sextic extensions $K\mathrm{/}\mathbb{Q}$ such that the Mordell-Weil group $E(K)$ has rank greater than the subgroup of $E(K)$ generated by all the $E(F)$ for the proper subfields $F\subset K$. For certain curves $E\mathrm{/}\mathbb{Q}$ we show that the number of such fields $K$ of conductor less than $X$ is $\gg \sqrt{X}$.