I'm trying to find basic facts known about the structure of class groups of binary quadratic forms with a discriminant d = -prime.
Wikipedia mentions the class groups (regardless of discriminant) are finite and abelian.
Then from the fundamental theorem of finite abelian groups, we know that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order.
Beyond this I am having trouble finding information.
Is the number of generators of a class group, with discriminant = -prime, unbounded?
Is there some restriction on what the decomposition looks like? Or is any combination of cyclic groups possible?
Using pari/gp to explore (using quadclassunit to calculate the size and structure of the group with discriminant -prime for many primes), it displays the structure as a cyclic decomposition with minimum number of generators. For instance:
? quadclassunit(-19919)
%1 = [135, [45, 3], [Qfb(2, 1, 2490), Qfb(45, -31, 116)], 1]
? quadclassunit(-3321607)
%2 = [567, [63, 3, 3], [Qfb(26, -19, 31942), Qfb(284, 259, 2983), Qfb(304, -107, 2741)], 1]
So $G(-19919) = C(45) \times C(3)$, and $G(-3321607) = C(63) \times C(3) \times C(3)$.
I've found some that require three generators, but I haven't found four or more yet. I assume that is just because I haven't searched for enough.
So far all the examples I've found of a product of three groups looks like this: $C(k)\times C(p^m) \times C(p^n)$ where $m,n \in \{1,2\}$, and $p$ prime. I've found over 400 so far, which I realize is a small number, but it seems a strange coincidence that the smaller factor groups always are based on the same prime.