On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number Fields

Consider the modular group PSL(2,Z)=⟨x,y|x2=y3=1⟩ generated by the transformations x:z↦−1/z and y:z↦(z−1)/z. Let H be the proper subgroup ⟨y,v|y3=v3=1⟩ of PSL(2,Z), where v=xyx. For a positive square-free integer n, this article studies the action of H on the subset {a+−nc|a,b=a2+nc,c∈Z,c≠0} of the imaginary quadratic number field Q(−n) where, in particular, the accurate estimate of the number of orbits arising from this action is given, correcting the estimate given in some of the relevant literature.