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If we know the prime factorisation of the fundamental negative discriminant $\Delta_K$, say $\Delta_K=p_1\cdot p_2 \cdots p_n$, then we are guaranteed that at least $2^{n-1}\mid h_K$, the class number of $C(\mathcal{O}_K)$.

Now, I would like to know the legendre conditions on the factors of $\Delta_K$ such that the size of the 2-Sylow subgroup of the class group is $2^{n-1}$. In other words, I want to guarantee that $2^{n-1}$ is the highest power of 2 that divides the $h_K$.

For example, when $\Delta_K=-pq$ and $\Delta_K\equiv1\pmod{4}$ then the condition $\left(\frac{p}{q}\right)=\left(\frac{q}{p}\right)=-1$ guarantees that the size of the 2-Sylow subgroup of the class group is exactly $2^{2-1}=2$. The example is from this paper which in turns refers to a French paper by Kaplan.

I asked this question first on the math site which yielded 0 response; so tried to ask here now.

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