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_p~- for all primes p ≡ 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.
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