I want to find the class number of $K=\mathbb{Q}(\sqrt{-23})$.
First I found Minkowski bound by using $n=2$, $s=1$ and the $disc(K)=-23$. It is bigger than $3$. So, it is enough to check for prime $p=2,3$. Then I got $<2>= P_{1}P_{2}$ and $<3>= Q_{1}Q_{2}$. Also, I showed that these ideals are not principal. But I am not sure about the next step.
There is a few examples here for calculation the class number: https://kconrad.math.uconn.edu/blurbs/gradnumthy/classgpex.pdf . These help me but there is no case $d<0$ and $d\equiv 1$ mod $4$ for $K=\mathbb{Q}(\sqrt{d})$.
I would be glad if you help me. Thanks.
