Determine the ideal class group of $\mathbf{Q}(\sqrt{-11})$.

Find all solutions of the diophantine equation

$$y^{2}+11=x^{3} \quad(x, y \in \mathbf{Z})$$

[Minkowski's bound is $n ! n^{-n}(4 / \pi)^{r_{2}}\left|D_{k}\right|^{1 / 2}$.]