  The periodic points of the algebraic function defined by the equation $g(x,y)
= x^3(4y^2+2y+1)-y(y^2-y+1)$ are shown to be expressible in terms of
Ramanujan's cubic continued fraction $c(\tau)$ with arguments in an imaginary
quadratic field in which the prime $3$ splits. If $w = (a+\sqrt{-d})/2$ lies in
an order of conductor $f$ in $K$ and $9 \mid N_{K/\mathbb{Q}}(w)$, then one of
these periodic points is $c(w/3)$, which is shown to generate the ring class
field of conductor $2f$ over $K$.
