Modular equations of a continued fraction of order six

Abstract

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 Vasuki et al. 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.