Let $\Lambda(\lambda) := \left\{\lambda, 1 - \lambda, \frac{1}{\lambda}, \frac{1}{1-\lambda}, \frac{\lambda - 1}{\lambda}, \frac{\lambda}{\lambda - 1} \right\}$, and consider the $j$-function
$j(\lambda) = 256\frac{(\lambda ^2 - \lambda + 1)^3}{\lambda ^2 (\lambda - 1)^2}.$
I'm trying to prove that \Lambda(\lambda) = \Lambda(\lambda ') iff j(\lambda) = j(\lambda ').
That the values of the $j$-function coincide if the sets are the same is fairly obvious by a straightforward calculation, but I'm stuck trying to prove the converse.
I'd welcome any hints.