Numerical Invariants of Totally Imaginary Quadratic-orders

LetAbe a real quadratic order of discriminantpor 4pwith a primep.In this paper we classify all proper totally imaginary quadraticA-ordersBwith indexω(B) = [B ×:A ×] > 1. We also calculate numerical invariants of these orders including the class number, the indexω(B) and the numbers of local optimal embeddings of these orders into quaternion orders. These numerical invariants are useful for computing the class numbers of totally definite quaternion algebras. 2010Mathematics Subject Classification. 11R52, 11G10. Key words and phrases.Class number formula, Arithmetic of quaternion algebras.