Some singular moduli for ${\bf Q}(\sqrt 3)$

In an earlier paper in this journal, the authors derived the equations which transform the Hilbert modular function field for ${\mathbf{Q}}(\sqrt 3 )$ when the arguments are multiplied by $(1 + \sqrt 3 ,1 - \sqrt 3 )$. These equations define a complex ${V_2}$, but we concentrate on special diagonal curves on which the values of some of the singular moduli can be evaluated numerically by using the "PSOS" algorithm. In this way the ring class fields can be evaluated for the forms ${\xi ^2} + {2^t}A{\eta ^2}$, where $A = 1,2,3,6$ and $t > 0$. These last results are based partly on conjectures supported here by numerical evidence.