On Ramanujan's cubic continued fraction

The periodic points of the algebraic function defined by the equation g(x,y)=x3(4y2+2y+1)-y(y2-y+1)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(x,y)=x<^>3(4y<^>2+2y+1)-y(y<^>2-y+1)=0$$\end{document} are shown to be expressible in terms of values of Ramanujan's cubic continued fraction c(tau)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(\tau )$$\end{document} with arguments in an imaginary quadratic field K in which the prime 3 splits. If w=(a+-d)/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w = (a+\sqrt{-d})/2$$\end{document} lies in an order of conductor f in K and 9 divided by NK/Q(w)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$9 \mid N_{K/\mathbb {Q}}(w)$$\end{document}, then one of these periodic points is c(w/3), which is shown to generate the ring class field of conductor 2f over K.