(*This summarizes the accepted answer of Somos, and uses just the j-function and Dedekind eta function*.) 

Given the [*j-function*][1] $j(\tau)$,

$$j(\tau) = 1728J(\tau)$$

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,

$$y_k = \frac{P_1(k)+\alpha P_2(k)}{\eta^2(\tau)}\\ \alpha=\frac{-1+\sqrt{-7}}2$$

for $k=1,2\dots7,$ with $P_1(k)$ and $P_2(k)$ as defined by Klein (and Somos).


  [1]: http://mathworld.wolfram.com/j-Function.html