  In this article we make an explicit approach to the higher degree case of the
problem: " For a given $CM$ field $M$, construct its maximal abelian extension
$C(M)$ (i.e. the Hilbert class field) by the adjunction of special values of
certain modular functions" in a restricted case. We make our argument based on
Shimura's main result on the complex multiplication theory of his article in
1967. His main result is constructed for a quaternion algebra $B$ over a
totally real number field $F$. We determine the modular function which gives
the canonical model for the case $B$ is coming from an arithmetic triangle
group. That is our main theorem. And we make an explicit case-study for $B$
corresponding to the triangle group $\Delta (3,3,5)$. The corresponding
canonical model appears as a restriction of the Appell's hypergeometric modular
function on a 2-dimensional hyperball to a hyperplane section. That is a
modular function for the family of the Koike pentagonal curves
$w^5=z(z-1)(z-\lambda_1)(z-\lambda_2)$ with two parameters
$\lambda_1,\lambda_2$. We use the result of K. Koike in 2003 to get an theta
representation of the canonical model function. By using this expression, we
show several examples of the Hilbert class fields of the $CM$ fields those are
embedded in the above $B$.
