Imaginary Bicyclic Biquadratic Number Fields with Class Number 5

An imaginary bicyclic biquadratic number field K is a field of the form ${\mathbb{Q}}({\sqrt{-m}},{\sqrt{-n}})$ where m and n are squarefree positive integers. The ideal class number h_K of K is the order of the abelian group I_K/P_K, where I_K and P_K are the groups of fractional and principal fractional ideals in the ring of integers 𝒪_K of K, respectively. This provides a measure on how far is 𝒪_K from being a PID. We determine all imaginary bicyclic biquadratic number fields with class number 5. We show there are exactly 243 such fields.