  Given an elliptic curve $E/\mathbb{Q}$, it is a conjecture of Goldfeld that
asymptotically half of its quadratic twists will have rank zero and half will
have rank one. Nevertheless, higher rank twists do occur: subject to the parity
conjecture, Gouv\^ea and Mazur constructed $X^{1/2-\epsilon}$ twists by
discriminants up to $X$ with rank at least two. For any $d\geq 3$, we build on
their work to consider twists by degree $d$ $S_d$-extensions of $\mathbb{Q}$
with discriminant up to $X$. We prove that there are at least
$X^{c_d-\epsilon}$ such twists with positive rank, where $c_d$ is a positive
constant that tends to $1/4$ as $d\to\infty$. Moreover, subject to a suitable
parity conjecture, we obtain the same result for twists with rank at least two.
