2-parts of class numbers of binary quadratic forms for non-fundamental discriminants

Question

I need a formula for the 2-adic valuation of the number of proper equivalence classes of primitive positive definite binary quadratic forms of discriminant $-D$, call it $h_0(-D)$. I'm sure the answer is well-known, and in fact I have found it on slides for a talk somewhere (see here, slide 21). According to this, the class number $h_0(-D)$ is precisely divisible by $2^{\mu-1}$ where $\mu$ can be given explicitly in terms of the number of odd prime divisors of $D$, depending on the congruence class of $-D\pmod{32}$. Does anyone have a reference for this which one could cite (the slides are not really such a reference I'd say)? The textbooks I know which address this subject only treat the case where $-D$ is a fundamental discriminant, and I think it requires a bit of work to arrive at the general case. So a reference would be much appreciated.

Thanks a lot in advance!

Answer 1

Pretty sure what you want is D. A. Buell, Binary Quadratic Forms. Chapter 9 is called The 2-Sylow Subgroup. Back in chapter 7, pages 117-118, he compares form class numbers $h(\Delta) $ and $h(\Delta p^2),$ also $h(4 \Delta)$

As indicated in comments, the number of genera is predictable (and is a power of 2). However, it is possible that the principal genus (and therefore each genus) has an even number of forms. On page 160, Buell refers to classes of order 4 in the group.

Not widely known, but I got some good use out of it, it was shown that the fourth powers in the class group coincide with the spinor kernel. See Estes and Pall Most of the pdfs I put at at KAP are about ternary forms.