 Ideal class group of the imaginary quadratic field of discriminant minus eighty-four

For $s=\sqrt{-21}$, the integer ring is $\mathbb{Z}[s]$. The Minkowski bound for ideal classes is less than six. The ramified primes $(2,1+s)$ and $(3,s)$ give independent classes of order two; their product is the class of $(5,s-2)$ because $(3+s)$ factors as the product of these three prime ideals. Element norms $a^2+21b^2$ cannot equal 2, 3 or 6, so these classes are distinct. Ideals of norm at most five supply no further classes, proving the four-element group.