(This summarizes the accepted answer of Somos, and uses just the j-function and Dedekind eta function.)
Given the j-function $j(\tau)$,
$$j(\tau) = 1728J(\tau)$$
implemented in Mathematica as j(t) = 1728KleinInvariantJ[t], then Klein's septic resolvent reduces to the elegant formula,
$$y\Big(y^3-\frac{8}{\alpha^3}\sqrt{-7}\Big)\Big(y^3-\sqrt{-7}\Big)=\sqrt[3]{j(\tau)}$$
where the seven roots are,
$$y_k = \frac{P_1(k)+\alpha P_2(k)}{\eta^2(\tau)}\\ \alpha=\frac{-1+\sqrt{-7}}2$$
with $P_1(k)$ and $P_2(k)$ for $k=1,2\dots7$ as defined by Klein (and Somos).