On Ramanujan’s cubic continued fraction

The periodic points of the algebraic function defined by the equation \(g(x,y)=x^3(4y^2+2y+1)-y(y^2-y+1)=0\) $g(x,y)=x^3(4y^2+2y+1)-y(y^2-y+1)=0$ are shown to be expressible in terms of values of Ramanujan’s cubic continued fraction \(c(\tau )\) $c(\tau )$ with arguments in an imaginary quadratic field K in which the prime 3 splits. If \(w = (a+\sqrt{-d})/2\) $w=(a+\sqrt{-d})/2$ lies in an order of conductor f in K and \(9 \mid N_{K/\mathbb {Q}}(w)\) $9\mid N_{K/\mathbb{Q}}(w)$ , then one of these periodic points is c(w/3), which is shown to generate the ring class field of conductor 2f over K.