A class number estimate

Question

Let $\mathcal{D} = \{D \in \mathbb{Z} : D \equiv 0, 1 \pmod{4}\}$ be the set of discriminants. It is well-known that each element in $\mathcal{D}$ is the discriminant of a primitive binary quadratic form, namely $x^2 - (D/4)y^2$ when $D \equiv 0 \pmod{4}$ and $x^2 + xy - (D-1)y^2/4$ when $D \equiv 1 \pmod{4}$.

For a primitive binary quadratic form $f$ the group $\text{SL}_2(\mathbb{Z})$ acts on it via substitution. Denote by $[f]$ the $\text{SL}_2(\mathbb{Z})$-equivalence class of $f$. Note that the discriminant is a $\text{SL}_2(\mathbb{Z})$-invariant, so that for each $g, h \in [f]$ we have $\Delta(g) = \Delta(h)$, so we may write $\Delta([f])$, the common value for $\Delta(g), g \in [f]$, to be the discriminant of the equivalence class. For each $D \in \mathcal{D}$, put

$$\displaystyle h(D) = \#\{[f] : f \text{ primitive}, \Delta([f]) = D\}$$

for the class number of primitive binary quadratic forms of discriminant $D$.

Observe that the set of squares is contained in $\mathcal{D}$, and it is well-known that $h(n^2) = \phi(n)$, and representatives of the distinct $\text{SL}_2(\mathbb{Z})$-classes of discriminant $n^2$ are given by

$$\displaystyle ax^2 + nxy : \gcd(a,n) = 1, 1 \leq a \leq n-1.$$

My question is, what about $h(-4n^2)$? Is there a nice characterization of them like the case of positive squares? Equivalently, can one find all reduced forms, i.e., a form $f(x,y) = ax^2 + bxy + cy^2$ with $|b| \leq a \leq c, a,c > 0$ of discriminant $-4n^2$?

Certainly, factoring $n^2 = u^2 v^2$ with $\gcd(u,v) = 1$ gives a primitive form $(ux)^2 + (vy)^2$ of discriminant $-4n^2$... but there are other forms such as $4x^2 + 4xy + 5y^2$ with discriminant $-64$ that do not arise this way.

Answer 1

This is from Buell, Binary Quadratic Forms, pages 109-119.

Note that $h(-16) = h(-4) = 1.$

When $p$ is an odd prime, we use the Legndre symbol in $$ h(-4p^2) = \frac{p - (-1|p)}{2} $$

After that, if $u > 1,$ while $u$ is allowed odd or even as needed; if $p$ does not divide $u,$ then $$ h(-4u^2p^2) = h(-4 u^2) \cdot \left(p - (-1|p) \right). $$ If $p$ divides $u,$ $$ h(-4u^2p^2) = h(-4 u^2) \cdot p. $$ $$ h(-16u^2) = 2 h(-4 u^2) $$

Let's see; note that $p - (-1|p)$ for an odd prime $p$ is always divisble by $4,$ so these class numbers are even, and the max power of $2$ grows. This is fair, the number of genera is growing with each new prime and is always a power of two, while each discriminant has a constant number of classes per genus.

For example, with odd $n > 1,$ if $$ n = \prod_p p^{e_p} $$ we get $$ h(-4n^2) = \frac{1}{2} \prod_p \left( p - (-1|p)\right) p^{e_p -1} $$

The conclusion is that $h(-4n^2)$ is mostly pretty similar in size to $\frac{n}{2} .$ In order to deviate much from that, it is most efficient to have $n$ squarefree odd, with either $n = 5 \cdot 13 \cdot 17 ...$ the product of the consecutive primes $1 \pmod 4$ up to some bound $P,$ or $n = 3 \cdot 7 \cdot 11 \cdot 19 ...$ the product of the consecutive primes $3 \pmod 4$ up to some bound. In these cases, the growth or shrinkage of $h$ is still bounded by Merten's Theorem, in one version (H+W_thm427) $\sum_{p \leq x} \frac{1}{p} = \log \log x + B + o(1).$ I guess if we are taking half the primes according to mod 4, we would expect the estimate to be halved.

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  n      -4 n^2      h      n factored 
  1          -4      1      factor n  =  1 
  2         -16      1      factor n  = 2
  3         -36      2      factor n  = 3
  4         -64      2      factor n  = 2^2
  5        -100      2      factor n  = 5
  6        -144      4      factor n  = 2 * 3
  7        -196      4      factor n  = 7
  8        -256      4      factor n  = 2^3
  9        -324      6      factor n  = 3^2
 10        -400      4      factor n  = 2 * 5
 11        -484      6      factor n  = 11
 12        -576      8      factor n  = 2^2 * 3
 13        -676      6      factor n  = 13
 14        -784      8      factor n  = 2 * 7
 15        -900      8      factor n  = 3 * 5
 16       -1024      8      factor n  = 2^4
 17       -1156      8      factor n  = 17
 18       -1296     12      factor n  = 2 * 3^2
 19       -1444     10      factor n  = 19
 20       -1600      8      factor n  = 2^2 * 5
 21       -1764     16      factor n  = 3 * 7
 22       -1936     12      factor n  = 2 * 11
 23       -2116     12      factor n  = 23
 24       -2304     16      factor n  = 2^3 * 3
 25       -2500     10      factor n  = 5^2
 26       -2704     12      factor n  = 2 * 13
 27       -2916     18      factor n  = 3^3
 28       -3136     16      factor n  = 2^2 * 7
 29       -3364     14      factor n  = 29
 30       -3600     16      factor n  = 2 * 3 * 5
 31       -3844     16      factor n  = 31
 32       -4096     16      factor n  = 2^5
 33       -4356     24      factor n  = 3 * 11
 34       -4624     16      factor n  = 2 * 17
 35       -4900     16      factor n  = 5 * 7
 36       -5184     24      factor n  = 2^2 * 3^2
 37       -5476     18      factor n  = 37
 38       -5776     20      factor n  = 2 * 19
 39       -6084     24      factor n  = 3 * 13
 40       -6400     16      factor n  = 2^3 * 5
 41       -6724     20      factor n  = 41
 42       -7056     32      factor n  = 2 * 3 * 7
 43       -7396     22      factor n  = 43
 44       -7744     24      factor n  = 2^2 * 11
 45       -8100     24      factor n  = 3^2 * 5
 46       -8464     24      factor n  = 2 * 23
 47       -8836     24      factor n  = 47
 48       -9216     32      factor n  = 2^4 * 3
 49       -9604     28      factor n  = 7^2
 50      -10000     20      factor n  = 2 * 5^2
 51      -10404     32      factor n  = 3 * 17
 52      -10816     24      factor n  = 2^2 * 13
 53      -11236     26      factor n  = 53
 54      -11664     36      factor n  = 2 * 3^3
 55      -12100     24      factor n  = 5 * 11
 56      -12544     32      factor n  = 2^3 * 7
 57      -12996     40      factor n  = 3 * 19
 58      -13456     28      factor n  = 2 * 29
 59      -13924     30      factor n  = 59
 60      -14400     32      factor n  = 2^2 * 3 * 5
 61      -14884     30      factor n  = 61
 62      -15376     32      factor n  = 2 * 31
 63      -15876     48      factor n  = 3^2 * 7
 64      -16384     32      factor n  = 2^6
 65      -16900     24      factor n  = 5 * 13
 66      -17424     48      factor n  = 2 * 3 * 11
 67      -17956     34      factor n  = 67
 68      -18496     32      factor n  = 2^2 * 17
 69      -19044     48      factor n  = 3 * 23
 70      -19600     32      factor n  = 2 * 5 * 7
 71      -20164     36      factor n  = 71
 72      -20736     48      factor n  = 2^3 * 3^2
 73      -21316     36      factor n  = 73
 74      -21904     36      factor n  = 2 * 37
 75      -22500     40      factor n  = 3 * 5^2
 76      -23104     40      factor n  = 2^2 * 19
 77      -23716     48      factor n  = 7 * 11
 78      -24336     48      factor n  = 2 * 3 * 13
 79      -24964     40      factor n  = 79
 80      -25600     32      factor n  = 2^4 * 5
 81      -26244     54      factor n  = 3^4
 82      -26896     40      factor n  = 2 * 41
 83      -27556     42      factor n  = 83
 84      -28224     64      factor n  = 2^2 * 3 * 7
 85      -28900     32      factor n  = 5 * 17
 86      -29584     44      factor n  = 2 * 43
 87      -30276     56      factor n  = 3 * 29
 88      -30976     48      factor n  = 2^3 * 11
 89      -31684     44      factor n  = 89
 90      -32400     48      factor n  = 2 * 3^2 * 5
 91      -33124     48      factor n  = 7 * 13
 92      -33856     48      factor n  = 2^2 * 23
 93      -34596     64      factor n  = 3 * 31
 94      -35344     48      factor n  = 2 * 47
 95      -36100     40      factor n  = 5 * 19
 96      -36864     64      factor n  = 2^5 * 3
 97      -37636     48      factor n  = 97
 98      -38416     56      factor n  = 2 * 7^2
 99      -39204     72      factor n  = 3^2 * 11
100      -40000     40      factor n  = 2^2 * 5^2
  n      -4 n^2      h      n factored 

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