Imaginary cyclic quartic fields with large minus class numbers

It is well-known that the minus class number h(p)(-) 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)(-) > p. In an attempt to find such an example, we have tabulated h; for all primes p equivalent to 5 (mod 8) with p < 10(10) and for primes p < 10(14) satisfying certain quartic character restrictions. An analysis of these data yields unconditional numerical evidence in support of the Cohen-Martinet heuristics, but as we did not find a value of p for which h(p)(-) > p by these methods, we constructed a 77-digit value of p for which one can prove h(p)(-) > p assuming the Extended Riemann Hypothesis.