Ranks of elliptic curves in cyclic sextic extensions of Q

For an elliptic curve E/Q we show that there are infinitely many cyclic sextic extensions K/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⊂K. For certain curves E/Q we show that the number of such fields K of conductor less than X is ≫X.