## Abstract

We study a continued fraction X(τ) of order six by using the modular function theory. We first prove the modularity of X(τ), and then we obtain the modular equation of X(τ) 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} (τ). Furthermore, we show that the value 1/X(τ) is an algebraic integer, and we present an explicit procedure for evaluating the values of X(τ) for infinitely many τ's in K.

Original language | English |
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Pages (from-to) | 202-219 |

Number of pages | 18 |

Journal | Open Mathematics |

Volume | 17 |

Issue number | 1 |

DOIs | |

State | Published - 2019 |

### Bibliographical note

Publisher Copyright:© 2019 Lee and Park, published by De Gruyter 2019.

## Keywords

- Ramanujan continued fraction
- modular equation
- modular function
- ray class fields