Let $K=\mathbb{Q}(\alpha)$ where $\alpha$ is a root of $X^{2}-X+12=0$. Factor the elements 2,3 , $\alpha$ and $\alpha+2$ as products of prime ideals in $\mathcal{O}_{K}$. Hence compute the class group of $K$.

Show that the equation $y^{2}+y=3\left(x^{5}-4\right)$ has no integer solutions.