If we know the prime factorisation of the fundamental negative discriminant $\Delta_K$, say $n$ prime factors, then we are guaranteed that $2^{n-1}\mid h_K$, the class number of $C(\mathcal{O}_K)$ that is $n-1$ is the 2 rank of the class group.

My question is how does one guarantee that there are no elements of order $4$ in the class group so that the 2-Sylow subgroup of the class group factors into $C(2)\ \times \cdots \times C(2)$ $n-1$ times. I know that there are some legendre symbol conditions that need to be satisfied on the prime factors of the discriminant but I don't know what the conditions are or how to get those conditions.


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