Some criteria on invariant values

Abstract

Let K = Q(root d) be a real quadratic field, where d is a positive square-free integer congruent to 1 modulo 4. In this paper we determine the first term of the symmetric part of the continued fraction expansion of the integral basis element omega(d) = (1 + root d)/2. Furthermore, we obtain a necessary and sufficient condition for Yokoi's d-invariant value to be zero and present some numerical results on the class number of such real quadratic fields.