Ideal class group of Q of square root minus thirty-one ($\mathrm{Cl}(\mathbb{Q}(\sqrt{-31}))\cong C_3$)

The ideal-form correspondence for imaginary quadratic fields and the reduction bound $a\le \sqrt{31/3}$ give exactly the forms $[1,1,8]$, $[2,1,4]$ and $[2,-1,4]$. Thus the class number is three and the ideal class group is cyclic of order three.