Summary: Let \(E_d(x)\) denote the “Euler polynomial” \(x^2+x+(1-d)/4\) if \(d\equiv 1\pmod 4\) and \(x^2-d\) if \(d\equiv 2,3\pmod 4\). Set \(\Omega(n)\) to be the number of prime factors (counting multiplicity) of the positive integer \(n\). The Ono invariant Ono\(_d\) of \(K=\mathbb Q(sqrt d)\) is defined to be \(\max\{\Omega(E_d(b)): b= 0,1,\dots,|\Delta_d|/4-1\) except when \(d=-1,-3\) in which case Ono\(_d\) is defined to be 1. Finally, let \(h_d=h_k\) denote the class number of \(K\). In 2002 J. Cohen and J. Sonn [J. Number Theory 95, No. 2, 259–267 (2002; Zbl 1082.11068)] conjectured that \(h_d=3\Leftrightarrow\text{Ono}_d=3\) and \(-d=p\equiv 3\pmod 4\) is a prime. They verified that the conjecture is true for \(p<1.5\times 10^7\). Moreover, they proved that the conjecture holds for \(p>10^{17}\) assuming the extended Riemann Hypothesis. In this paper, we show that the conjecture holds for \(p\leq 2.5\times 10^{13}\) by the aid of computer. And using a result of Bach, we also prove that the conjecture holds for \(p>2.5\times 10^{13}\) assuming the extended Riemann Hypothesis. In conclusion, we prove the conjecture is true assuming the extended Riemann Hypothesis.