  Let $N$ be a square-free integer such that the arithmetic group
$\Gamma_0(N)^+$ has genus zero; there are $44$ such groups. Let $j_N$ denote
the associated Hauptmodul normalized to have residue equal to one and constant
term equal to zero in its $q$-expansion. In this article we prove that the
Hauptmodul at any elliptic point of the surface associated to $\Gamma_0(N)^+$
is an algebraic integer. Moreover, for each such $N$ and elliptic point $e$, we
show how to explicitly evaluate $j_{N}(e)$ and provide the list of generating
polynomials (with small coefficients) of the class fields or their subfields
corresponding to the orders over the imaginary quadratic extension of rationals
stemming from the elliptic points under consideration.
