Math 358 Fundamental discriminants of positive definite quadratic forms

Number.	D = Fundamental discriminant	h(D)=class number	List of all reduced forms	Structure of the class group	representatives of ideal classes
1.	-3	1	[1,1,1]	trivial
2.	-4	1	[1,0,1]	trivial
3.	-7	1	[1,1,2]	trivial
4.	-8	1	[1,0,2]	trivial
5.	-11	1	[1,1,3]	trivial
6.	-15	2	[1,1,4],[2,1,2]	Z/2
7.	-19	1	[1,1,5]	trivial
8.	-20	2	[1,0,5],[2,2,3]	Z/2
9.	-23	3	[2,1,3],[2,-1,3],[1,1,6]	Z/3	(1),((1+sqrt(-23))/2, 3), ((1+sqrt(-23))/2+2, 3)
10.	-24	2	[1,0,6],[2,0,3]	Z/2
11.	-31	3	1,1,8],[2,1,4],[2,-1,4]	Z/3	(1),(2,(1+Sqrt[-31])/2),(2,(-1+Sqrt[-31])/2)
12.	-35	2	[1,1,9],[3,1,3]	Z/2	(1,),(2,(-35)^(1/2) + 1)
13.	-39	4	[1,1,10],[2,-1,5],[2,1,5],[3,3,4]	Z/4	(1),(2,(1+Sqrt(-39))/2),(2,(1-Sqrt(-39))/2),(3,(3-Sqrt(-39))/2)
14.	-40	2	[1,0,10],[2,0,5]	Z/2	(1),(2, sqrt(-10))
15.	-43	1	[1,1,11]	trivial
16.	-47	5	[1,1,12],[2,+-1,6],[3,+-1,4]	Z/5
17.	-51	2	[1,1,13], [3,3,5]	Z/2
18.	-52	2	[1,0,13],[2,2,7]	Z/2
19.	-55	4	[2,-1,7],[2,1,7],[4,3,4],[1,1,14]	Z/4
20.	-56	4	[1,0,14],[2,0,7],[3,+-2,5]	Z/4
21.	-59	3	[1,1,15],[3,+_1,5]	Z/3
22.	-67	1	[1,1,17]	trivial
23.	-68	4	[1,0,17],[2,2,9],[3,-2,6],[3,2,6]	Z/4
24.	-71	7	[1,1,18],[2,1,9],[2,-1,9],[3,1,6],[3,-1,6],[4,3,5],[4,-3,5]	Z/7
25.	-79	5	[1,1,20], [2,+-1,10], [4,+-1,5]	Z/5
26.	-83	.	.	.
27.	-84	4	[1,0,21],[2,2,11],[3,0,7],[5,4,5]	Z/2 x Z/2
28.	-87	6	[1,1,22], [2, +-1, 11], [3,3,8], [4,+-3, 6]	Z/6
29.	-88	2	[13,-4,-2],[1,0,22]	Z/2
30.	-91	2	[1,1,23],[5,3,5]	Z/2
31.	-95	8	[1,1,24],[2,+_1,12],[3,+_1,8],[4,+_1,6],[5,5,6]	Z/8
32.	-103	5	[1,1,26],[2,-1,13],[2,1,13],[4,-3,7],[4,3,7]	Z/5
33.	-104	6	[1,0,26],[2,0,13],[3,-2,9],[3,2,9],[5,-4,6],[5,4,6]	Z/6
34.	-107	3	[1,1,27],[3,1,9],[3,-1,9]	Z/3
35.	-111	8	[1,1,28], [2,+-1,14], [3,3,10], [4,+-1,7], [5,+-3,6]	Z/8
36.	-115	2	[1,1,29],[5,5,7]	Z/2
37.	-116	6	[1, 0, 24], [2, 2, 15], [3, +-2, 10], [5, +-2, 6]	Z/6
38.	-119	10	[1, 1, 30], [2, +-1, 15], [3, +-1, 10], [4, +-3, 8], [5,+-1, 6], [6, 5, 6]	Z/10
39.	-120	4	[1,0,30],[2,0,15],[3,0,10],[5,0,6]	Z/2 x Z/2
40.	-123	2	[1,1,31],[3,3,11]	Z/2
41.	-127	5	[1,1,32],[2,+_1,16],[4,+_1,8]	Z/5
42.	-131	5	[1,1,33],[3,-1,11],[3,1,11],[5,-3,7],[5,3,7]	Z/5
43.	-132	4	[1,0,33],[2,2,17],[3,0,11],[6,6,7]	Z/2 x Z/2
44.	-136	4	[1,0,34], [2,0,17],[5,2,7],[5,-2,7]	Z/4
45.	-139	3	[1,1,35], [5,+-1,7]	Z/3
46.	-143	10	[1,1,36],[6,5,7],[6,-5,7],[2,1,18],[2,-1,18],[3,1,12],[3,-1,12], [4,1, 9],[4,-1,9][6,1,6]	Z/10
47.	-148	2	[1, 0, 37], [2, 2, 19]	Z/2
48.	-151	7	[1, 1, 38], [2, +-1, 19], [4, +-3, 10], [5, +-3, 8]	Z/7
49.	-152	6	[3,2,13],[6,4,7],[2,0,19],[6,-4,7],[3,-2,13],[1,0,38]	Z/6
50.	-155	4	[1,1,39],[3,+-1,13],[5,5,9]	Z/4
51.	-159	10	[1,1,40],[2,+_1,20],[3,3,14],[4,+_1,10],[5,+_1,8],[6,+_3,7]	Z/10
52.	-163	1	[1,1,41]	trivial
53.	-164	8	[1,0,41],[2,2,21],[3,-2,14],[3,2,14],[5,-4,9],[5,4,9],[6,-2,7],[6,2,7]	Z/8
54.	-167	11	[1,1,42],[2,1,21],[3,1,14],[6,1,7],[4,3,11],[6,5,8],[2,-1,21],[3,-1,14],[6,-1,7],[4,-3,11],[6,-5,8]	Z/11
55.	-168	4	[1,0,42], [2,0,21], [3,0,14], [6,0,7]	Z/2 x Z/2
56.	-179	.	.	.
57.	-183	8	[1, 1, 46], [2, +-1, 23], [3, 3, 16], [4, +-3, 12], [6, +-3, 8]	Z/8
58.	-184	4	[1,0,46], [2, 0, 23], [5, +-4, 10]	Z/4
59.	-187	2	[7,3,7],[1,1,47]	Z/2
60.	-191	13	[1,1,48],[2,+_1,24],[3,+_1,16],[4,+_ 1,12],[6,+_1,8],[5,+_3,10],[6,+_5,9]	Z/13
61.	-195	4	[1,1,49],[3,3,17],[5,5,11],[7,1,7]	Z/2 x Z/2
62.	-199	9	[1,1,50],[2,-1,25],[2,1,25],[4,-3,13],[4,3,13],[5,-1,10],[5,1,10],[7,-5,8],[7,5,8]	Z/9
63.	-203	4	[1,1,51],[3,-1,17],[3,1,17],[7,7,9]	Z/4
64.	-211	3	[1,1,53],[5,3,11],[5-3,11]	Z/3
65.	-212	6	[1,0,53], [2,2,27], [3,+-2,18], [6,+-2,9]	Z/6
66.	-215	.	.	.
.	.	.	.	.
.	-235	2	[1,1,59],[5,5,13]	Z/2

Discriminants of real quadratic fields

We list below the first few positive fundamental discriminants. (These are the numbers of the form
d with d square free and congruent to 1 mod 4 and 4d with d square free and congruent to 2 or 3 mod 4.)

Number.	D = Fundamental discriminant	h(D)=class number	h^+(D)=# of narrow ideal classes	Cycle structure of reduced forms (; separates cycles)	Unit of norm -1	Unit of norm 1	decomposition of primes: 1=split,0=ramified,-1=inert	2	3	5	7	11	13	17	19	23	29
1.	5	1	1	[1,1,-1],[-1,1,1]	(1+sq(5))/2	(3+sq(5))/2	.	-1	-1	0	-1	1	-1	-1	1	-1	1
2.	8	1	1	[1,2,-1],[-1,2,1]	1+sq(2)	3+2sq(2)	.	0	-1	-1	1	-1	-1	1	-1	1	-1
3.	12	1	2	[1,2,-2],[-2,2,1];[-1,2,2],[2,2,-1]	None	2+sq(3)	.	0	0	-1	-1	1	1	-1	-1	1	-1
4.	13	1	1	[1,3,-1],[-1,3,1]	(3+sq(13))/2	(11+3sq(13))/2	.	.	.	.	.	.	.	.	.	.	.
5.	17	1	1	[2,1,-2],[-2,3,1],[1,3,-2],[-2,1,2],[2,3,-1],[-1,3,2]	4+sq(17)	33+8sq(17)	.	1	0	0	-1	-1	-1	-1	1	1	-1
6.	21	1	2	.	None	(5+sqr(21))/2	.	-1	0	1	0	-1	-1	1	-1	-1	-1
7.	24	1	2	[1, 4, -2], [-2, 4, 1];[-1, 4, 2], [2, 4, -1]	None	5 + 2*sqrt(6)	.	0	0	1	-1	-1	-1	-1	1	1	1
8.	28	1	2	.	None	8+sqr(7)3	.	0	-1	1	0	-1	-1	-1	1	-1	1
9.	29	1	.	.	.	.	.	-1	-1	1	1	-1	1	-1	-1	1	0
10.	33	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.
11.	37	1	1	[3,1,-3],[-3,5,1],[1,5,-3],[-3,1,3],[3,5,-1],[-1,5,3]	6+Sqrt[37]	73+12*Sqrt[37]	.	.	.	.	.	.	.	.	.	.	.
12.	40	2	2	[3,2,-3],[-3,4,2],[2,4,-3],[-3,2,3],[3,4,-2],[-2,4,3];[1,6,-1],[-1,6,1]	-3-sqrt(10)	19+3sqrt(40)	.	.	.	.	.	.	.	.	.	.	.
13.	41	1	1	[-4, 3, 2],[-4, 5, 1],[-2, 3, 4],[-2, 5, 2],[-1, 5, 4],[1, 5, -4],[2, 3, -4],[2, 5, -2],[4, 3, -2],[4, 5, -1]	32+5sq(41)	2049+320sq(41)	.	.	.	.	.	.	.	.	.	.	.
14.	44	1	2	[1,6,-2],[-2,6,1]; [-1,6,2],[2,6,-1]	None	10 + 3sq(11)	.	.	.	.	.	.	.	.	.	.	.
15.	53	1	1	[1,7,-1],[-1,7,1]	(7+sq(53))/2	51+7sq(53))/2	.	.	.	.	.	.	.	.	.	.	.
16.	56	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.
17.	57	1	2	[3, 3, -4], [-4, 5, 2], [2, 7, -1], [-1, 7, 2], [2, 5, 4], [-4, 3, 3];[-3, 3, 4], [4, 5, -2], [-2, 7, 1], [1, 7, -2], [-2, 5, 4], [4, 3, -3]	None	(302+40*sqrt(57))/2	.	.	.	.	.	.	.	.	.	.	.
18.	60	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.
19.	61	1	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.
20.	65	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.
21.	69	1	2	[3,3,-5],[-5,7,1],[1,7,-5],[-5,3,3];[-3,3,5],[5,7,-1],[-1,7,5],[5,3,-3]	None	(25+3*Sqrt[69])/2	.	.	.	.	.	.	.	.	.	.	.
22.	73	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.
23.	76	1	2	[-5, 4, 3],[-5, 6, 2],[-1, 8, 3],[2, 6, -5],[3, 4, -5],[3, 8,-1];[-3, 4, 5],[-3, 8, 1],[-2, 6, 5],[1, 8, -3],[5, 4, -3],[5, 6, -2]	None	170 + 39sq(19)	.	.	.	.	.	.	.	.	.	.	.
24.	77	1	2	[1,7,-7],[-7,7,1];[-1,7,7],[7,7,-1]	None	(9 + sq(77))/2	.	.	.	.	.	.	.	.	.	.	.
25.	85	2	2	[1,9,-1],[-1,9,1]; [3,7,-3],[-3,5,5],[5,5,-3],[-3,7,3],[3,5,-5],[-5,5,3]	(9+sq(85))/2	(83+9sq(85))/2	.	.	.	.	.	.	.	.	.	.	.

Non-totally-real cubic fields, L, of small discriminant

Notation for decomposition of primes: A prime of L over a rational prime p will be
denoted P(or P' or P"),q,Q according to degree of the residue class field extension being 1,2,3.

Number D = Discriminant Min. poly. f(x) of a primitive element Integers class number Galois group Associated quadratic field decomposition of primes 2 3 5 7 11 13 17 19 23 29

. -23 x^3 - x + 1 Z[x]/(f) 1 S_3 Q(sqrt(D)) . Q Q Pq Pq Pq Q Pq Pq P'P^2 Q

. -31 x^3+x+1 Z[x]/(f) 1 S_3 Q(sqrt(D)) . . . . . . . . . . .

. -31 x^3-2x^2+x+1 Z[x]/(f) 1 S_3 Q(sqrt(D)) . Q Pq Q Q Pq Pq Pq Q Pq Pq

. -44 x^3 + 2x^2 - 2 Z[x]/(f) 1 S_3 Q(sqrt(D)) . P^3 Pq Q Q P'P^2 Pq Pq Pq Q Pq

. -59 x^3 + 2x + 1 Z[x]/(f) 1 S_3 Q(sqrt(D)) . Pq Q Q Q Pq Pq PP'P" Q Pq Q

. -76 x^3 + 3x^2+x+1 Z[x]/(f) 1 S_3 Q(sqrt(D)) . P^3 Pq Q Q Q Pq Q P'P^2 PP'P" Pq

. -83 x^3-x^2+x-2 Z[x]/(f) 1 S_3 Q(sqrt(D)) . Pq Q Pq Q Q Pq Q Pq PP'P" Q

. -104 x^3-x+2 Z[x]/(f) 1 S_3 Q(sqrt(D) . . . . . . . . . . .

The two fields with discriminant -31 are isomorphic.