  For an elliptic curve $E/\mathbb{Q}$ we show that there are infinitely many
cyclic sextic extensions $K/\mathbb{Q}$ 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 $E/\mathbb{Q}$ we show
that the number of such fields $K$ of conductor less than $X$ is $\gg\sqrt X$.
