I am reading the paper "AN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION" by Kathrin Bringmann and Ken Ono. In the proof of Proposition 4.2 they make reference to the existence of a bijection between the solutions to the Diophantine equation with $k,c>0$

$b^2-24kc=1-24n$

and the points of the orbits

$\displaystyle \bigcup_{Q\in Q_{24n-1}^p/\Gamma_0(6)}\lbrace{ A\tau_{Q}:A\in \Gamma_0(6)/\Gamma_{\tau_Q}\rbrace}.$

They do give some reasons why this exists but I am having trouble filling in the details. Does anyone know how to fill in the details that I think I am missing?