Summary: We study a continued fraction \(X(\tau)\) of order six by using the modular function theory. We first prove the modularity of \(X(\tau)\), and then we obtain the modular equation of \(X(\tau)\) of level \(n\) for any positive integer \(n\); this includes the result of K. R. Vasuki et al. [Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 56, No. 1, 77–89 (2010; Zbl 1205.11013)] for \(n=2,3,5,7\) and 11. As examples, we present the explicit modular equation of level \(p\) for all primes \(p\) less than 19. We also prove that the ray class field modulo 6 over an imaginary quadratic field \(K\) can be obtained by the value \(X^2(\tau)\). Furthermore, we show that the value \(1/X(\tau)\) is an algebraic integer, and we present an explicit procedure for evaluating the values of \(X(\tau)\) for infinitely many \(\tau\)’s in \(K\).