The following is my efforts on this question.
Let $K$ be the number field $\mathbb{Q}(\sqrt{-3},\sqrt{5})$, and $\mathcal{O}_K$ be the ring of integers of $K$. Let $k=-3\cdot5/(-3,5)^2=-15$. Since $-3\equiv5\equiv-15\equiv1(\text{ mod 4})$, then the discriminant $d(K)$ of $K$ is $-3\cdot5\cdot(-15)=15^2$. Then I worked out the Minkowski bound of $K$ is $45/2\pi^2<2.28$, and hence it suffices to consider the prime ideals of $\mathcal{O}_K$ over $2$.
I noticed that $\mathbb{Q}(\sqrt{-15})\subseteq\mathbb{Q}(\sqrt{-3},\sqrt{5})\subseteq\mathbb{Q}(e^{\frac{2\pi i}{15}})$. Since $-15\equiv1(\text{mod 4})$, the discriminant of $\mathbb{Q}(\sqrt{-15})$ is $-15$. Then $-15\equiv1(\text{mod 8})$ implies that $2$ splits completely in $\mathbb{Q}(\sqrt{-15})$. But $2$ splits completely in $\mathbb{Q}(e^{\frac{2\pi i}{15}})$, since the minimal polynomial $f(x)$ of $e^{\frac{2\pi i}{15}}$ over $\mathbb{Q}$ satisfies that $f(x)\equiv(x^4+x+1)(x^4+x^3+1)(\text{mod 2})$. Then the Galois theory shows that $2$ splits into two distnct prime ideals in $\mathbb{Q}(\sqrt{-3},\sqrt{5})$, say $2\mathcal{O}_K=\mathfrak{p}_{1}\mathfrak{p}_{2}$. But I have no idea to show whether $\mathfrak{p}_{1}$(or $\mathfrak{p}_{2}$) is a principal ideal in $\mathcal{O}_K$ or not.
Plus: Since the discriminants of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(\sqrt{5})$ are $-3$ and $5$, respectively. Also, $K$ is the composite field of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(\sqrt{5})$. Then $\mathcal{O}_K=\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}\mathcal{O}_{\mathbb{Q}(\sqrt{5})}$.
I'd appreciate your help!