Imaginary Cyclic Quartic Fields with Large Minus Class Numbers

It is well-known that the minus class number h \(_{p}^{\rm -}\) of an imaginary cyclic quartic field of prime conductor p can grow arbitrarily large, but until now no one has been able to exhibit an example for which h \(_{p}^{\rm -}\) > p. In an attempt to find such an example, we have tabulated h \(_{p}^{\rm -}\) for all primes p ≡ 5(mod 8) with p p by these methods, we constructed a 77-digit value of p for which one can prove h \(_{p}^{\rm -}\) > p assuming the Extended Riemann Hypothesis.