I wish to compute class group of $K = \mathbb{Q}(\sqrt{30})$. I have checked, using sage, that the group is $C_2$, but I have the incorrect answer of $C_2\times C_2$. I outline my line of reasoning below; I would appreciate some guidance on how to correct my approach. To save some reading, I seem to be making a mistake when it comes to checking if $P_2P_5$ is principal or not.
The Minkowski bound is $\sqrt30$, and so I've checked how the rational primes $2$,$3$ and $5$ factorise in $K$, with $\alpha = \sqrt30$
$$(2) = (2,\alpha)^2 = P_2^2$$ $$(3) = (3,\alpha)^2 = P_3^2$$ $$(5) = (5,\alpha)^2 = P_5^2$$
I have checked that these prime ideals are not principal. We see that the orders of all the elements in the class group are $2$, and $P_2P_3=(6+\alpha)$, so $P_2$ and $P_3$ are in the same ideal class. Since I should be getting a class group of order 2, I expect something similar to happen when computing $P_2P_5$, yet if this were to be principal, generated by $x+y\alpha$, say, then, since $N(P_2P_5) = 10$, we need integer solutions to: $$x^2 - 30y^2 = \pm10$$ A number theoretic argument, or a check on Wolfram Alpha, shows that no such solutions can be found. This has led me to believe that $P_2$ and $P_5$ are not in the same ideal class (else, since the order of the ideal class is $2$, the product of the ideals would be principal), and hence that the class group is $C_2\times C_2$.
