Class Numbers

Proposition 2.1   Let $D<0$ be a discriminant. There are only finitely many equivalence classes of positive definite binary quadratic forms of discriminant $D$.

Proof. Since there is exactly one reduced binary quadratic form in each equivalence class, it suffices to show that there are only finitely many reduced forms of discriminant $D$. Recall that if a form $(a,b,c)$ is reduced, then $\vert b\vert \leq a \leq c$. If $(a,b,c)$ has discriminant $D$ then $b^2-4ac=D$. Since $b^2\leq a^2 \leq ac$, we have $D = b^2-4ac\leq -3ac$, so

$$3ac \leq -D.$$

There are only finitely many positive integers $a,c$ that satisfy this inequality. $\qedsymbol$

Definition 2.2   A binary quadratic form $(a,b,c)$ is primitive if $\gcd(a,b,c)=1$.

Definition 2.3   The class number $h_D$ of discriminant $D<0$ is the number of equivalence classes of primitive positive definite binary quadratic forms of discriminant $D$.

I computed the following table of class number $h_D$ for $-D\leq 839$ using the built-in PARI function qfbclassno(D,1). Notice that there are just a few $1$s at the beginning and then no more.

$-D$	$h_{D}$
3	1
7	1
11	1
15	2
19	1
23	3
27	1
31	3
35	2
39	4
43	1
47	5
51	2
55	4
59	3
63	4
67	1
71	7
75	2
79	5
83	3
87	6
91	2
95	8
99	2
103	5
107	3
111	8
115	2
119	10

$-D$	$h_{D}$
123	2
127	5
131	5
135	6
139	3
143	10
147	2
151	7
155	4
159	10
163	1
167	11
171	4
175	6
179	5
183	8
187	2
191	13
195	4
199	9
203	4
207	6
211	3
215	14
219	4
223	7
227	5
231	12
235	2
239	15

$-D$	$h_{D}$
243	3
247	6
251	7
255	12
259	4
263	13
267	2
271	11
275	4
279	12
283	3
287	14
291	4
295	8
299	8
303	10
307	3
311	19
315	4
319	10
323	4
327	12
331	3
335	18
339	6
343	7
347	5
351	12
355	4
359	19

$-D$	$h_{D}$
363	4
367	9
371	8
375	10
379	3
383	17
387	4
391	14
395	8
399	16
403	2
407	16
411	6
415	10
419	9
423	10
427	2
431	21
435	4
439	15
443	5
447	14
451	6
455	20
459	6
463	7
467	7
471	16
475	4
479	25

$-D$	$h_{D}$
483	4
487	7
491	9
495	16
499	3
503	21
507	4
511	14
515	6
519	18
523	5
527	18
531	6
535	14
539	8
543	12
547	3
551	26
555	4
559	16
563	9
567	12
571	5
575	18
579	8
583	8
587	7
591	22
595	4
599	25

$-D$	$h_{D}$
603	4
607	13
611	10
615	20
619	5
623	22
627	4
631	13
635	10
639	14
643	3
647	23
651	8
655	12
659	11
663	16
667	4
671	30
675	6
679	18
683	5
687	12
691	5
695	24
699	10
703	14
707	6
711	20
715	4
719	31

$-D$	$h_{D}$
723	4
727	13
731	12
735	16
739	5
743	21
747	6
751	15
755	12
759	24
763	4
767	22
771	6
775	12
779	10
783	18
787	5
791	32
795	4
799	16
803	10
807	14
811	7
815	30
819	8
823	9
827	7
831	28
835	6
839	33

We can compute these numbers using Proposition 2.1. The following PARI program enumerates the primitive reduced forms of discriminant $D$.

{isreduced(a,b,c) = 
   if(b^2-4*a*c>=0 || a<0, 
      error("reduce: (a,b,c) must be positive definite."));
   if(!(abs(b)<=a && a<=c), return(0));
   if(abs(b)==a || a==c, return(b>=0));
   return(1);
}
{reduce(f) = 
   local(D, k, t, a,b,c); 
   a=f[1]; b=f[2]; c=f[3]; D=b^2-4*a*c;
   if(D>=0 || a<0, error("reduce: (a,b,c) must be positive definite."));
   while(!isreduced(a,b,c),       \\ ! means ``not''
      if(c<a, 
         b = -b; t = a; a = c; c = t,
      \\ else
         if (abs(b)>a || -b==a, 
            k = floor((a-b)/(2*a)); 
            b = b+2*k*a;
            c = (b^2-D)/(4*a);
         )
      )
   );
   return([a,b,c])
}
{reducedforms(D)=
   local(bound, forms, b, r);
   if (D > 0 || D%4 == 2 || D%4==3, error("Invalid discriminant"));
   bound = floor(-D/3);
   forms = [];
   for(a = 1, bound,
      for(c = 1, bound,
         if(3*a*c<=-D && issquare(4*a*c+D),
            b = floor(sqrt(4*a*c+D));
            r = reduce([a,b,c]);
            print1([a,b,c], " ----> ", r);
            if (gcd(r[1],gcd(r[2],r[3])) == 1,
               forms = setunion(forms,[r]); print(""),
               \\ else
               print ("  \t(not primitive)")
            )
         )
      )
   );
   return(eval(forms));    \\ eval gets rid of the annoying quotes.
}

For example, when $D=-419$ the program finds exactly $9$ reduced forms:

? D = -419
%21 = -419
? qfbclassno(D,1)
%22 = 9
? reducedforms(D)
[1, 1, 105] ----> [1, 1, 105]
[1, 3, 107] ----> [1, 1, 105]
[1, 5, 111] ----> [1, 1, 105]
[1, 7, 117] ----> [1, 1, 105]
[1, 9, 125] ----> [1, 1, 105]
[1, 11, 135] ----> [1, 1, 105]
[3, 1, 35] ----> [3, 1, 35]
[3, 5, 37] ----> [3, -1, 35]
[3, 7, 39] ----> [3, 1, 35]
[3, 11, 45] ----> [3, -1, 35]
[5, 1, 21] ----> [5, 1, 21]
[5, 9, 25] ----> [5, -1, 21]
[5, 11, 27] ----> [5, 1, 21]
[7, 1, 15] ----> [7, 1, 15]
[9, 7, 13] ----> [9, 7, 13]
[9, 11, 15] ----> [9, -7, 13]
[13, 7, 9] ----> [9, -7, 13]
[15, 1, 7] ----> [7, -1, 15]
[15, 11, 9] ----> [9, 7, 13]
[21, 1, 5] ----> [5, -1, 21]
[25, 9, 5] ----> [5, 1, 21]
[27, 11, 5] ----> [5, -1, 21]
[35, 1, 3] ----> [3, -1, 35]
[37, 5, 3] ----> [3, 1, 35]
[39, 7, 3] ----> [3, -1, 35]
[45, 11, 3] ----> [3, 1, 35]
[105, 1, 1] ----> [1, 1, 105]
[107, 3, 1] ----> [1, 1, 105]
[111, 5, 1] ----> [1, 1, 105]
[117, 7, 1] ----> [1, 1, 105]
[125, 9, 1] ----> [1, 1, 105]
[135, 11, 1] ----> [1, 1, 105]
%23 = [[1, 1, 105], [3, -1, 35], [3, 1, 35], [5, -1, 21], [5, 1, 21], 
       [7, -1, 15], [7, 1, 15], [9, -7, 13], [9, 7, 13]]
? length(%23)
%24 = 9

Theorem 2.4 (Heegner, Stark-Baker, Goldfeld-Gross-Zagier)   Suppose $D$ is a negative discriminant that is either square free or $4$ times a square-free number. Then

• $h_D=1$ only for $D=-3,-4,-7,-8,-11,-19,-43,-67,-163$.
• $h_D=2$ only for $D=-15, -20, -24, -35, -40, -51, -52, -88, -91,$
  $-115,-123, -148, -187, -232, -235, -267, -403, -427$.
• $h_D=3$ only for $D=-23,-31,-59,-83,-107,-139,-211,-283,-307,$
  $-331,-379,-499,-547,-643,-883,-907.$
• $h_D=4$ only for $D=-39,-55,-56,-68,\ldots,-1555.$

To quote Henri Cohen: ``The first two statements concerning class numbers $1$ and $2$ are very difficult theorems proved in 1952 by Heegner and in 1968-1970 by Stark and Baker. The general problem of determing all imaginary quadratic fields with a given class number has been solved in principle by Goldfeld-Gross-Zagier, but to my knowledge the explicit computations have been carried to the end only for class numbers $3$ and $4$ (in addition to the already known class numbers $1$ and $2$).