  Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of
the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the
assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a
divisor of $8$. We compute all $D$ with $|D|\leq 3.1\cdot 10^{20}$ such that
$E(D)\leq 8$.
