-degree\] Let $E/\mathbb{Q}$ be an elliptic curve let $d\geq 0$ be an integer. There is a constant $c_d>0$ such that for each $\varepsilon =\pm 0$, there are $\gg X^{c_d - \epsilon}$ fields $K\in\mathcal{F}_d(X)$ with $w(E,\rho_K)=\varepsilon$ and $\mathrm{rk}(E(K)) > \mathrm{rk}(E(\mathbb{Q}))$. 