Class Numbers¶

mdsage.quadratic_class_numbers.class_number(D)¶

INPUT:

D - an negative integer that is 0 or 1 mod 4

OUTPUT:

An integer that is the classnumber of the quadratic order of discriminant D

EXAMPLES:

sage: from mdsage import *
sage: class_number(-4)
1
sage: class_number(-47)
5
sage: class_number(-7*31**2)
32
sage: class_number(-7*29**2)
28

see http://www.math.toronto.edu/~ila/Cox-Primes_of_the_form_x2+ny2.pdf could maybe use theorem 7.24 to speed this up

mdsage.quadratic_class_numbers.class_numbers(D, fs, clnr_D=None)¶

computes the class number h(D*f**2) for a bunch of f’s at the same time using theorem 7.24 from http://www.math.toronto.edu/~ila/Cox-Primes_of_the_form_x2+ny2.pdf

INPUT:

D - an negative integer that 0 or 1 mod 4
fs - a list of integers
clnr_D (optional) - an integer that equals class_number(D) to speed up the computation

OUTPUT:

a list of tuples of integers equalling [(f,class_number(D*f**2)) for f in fs]

EXAMPLES:

sage: from mdsage import *
sage: list(class_numbers(-7, [1, 29, 31]))
[(1, 1), (29, 28), (31, 32)]

mdsage.quadratic_class_numbers.cm_orders2(bound)¶

yields all triples (h(D*f**2),D,f) such that h(D*f**2) <= bound

INPUT:

bound - an integer up to which to generate the cm_orders

OUTPUT:

an iterator over all triples (h(D*f**2),D,f) such that h(D*f**2) <= bound

EXAMPLES:

sage: from mdsage import *
sage: list(cm_orders2(1))
[(1, -3, 1),
 (1, -3, 2),
 (1, -3, 3),
 (1, -4, 1),
 (1, -4, 2),
 (1, -7, 1),
 (1, -7, 2),
 (1, -8, 1),
 (1, -11, 1),
 (1, -19, 1),
 (1, -43, 1),
 (1, -67, 1),
 (1, -163, 1)]

mdsage.quadratic_class_numbers.small_class_number_discriminants(bound=100)¶

returns a dictionary of all imaginary quadratic fundamental discriminants of small class number

INPUT:

bound (optional) - an integer up to which class number to compute the fundamental discriminants

OUTPUT:

a dictionary whose keys are class numbers and whose values are lists of discriminants having that class number

EXAMPLES:

sage: from mdsage import *
sage: small_class_number_discriminants(2)
{1: [-3, -12, -27, -4, -16, -7, -28, -8, -11, -19, -43, -67, -163],
 2: [-48,
  -75,
  -147,
  -36,
  -64,
  -100,
  -112,
  -32,
  -72,
  -99,
  -15,
  -60,
  -20,
  -24,
  -35,
  -40,
  -51,
  -52,
  -88,
  -91,
  -115,
  -123,
  -148,
  -187,
  -232,
  -235,
  -267,
  -403,
  -427]}
sage: # if the following fails the data in our article is also incorrect
sage: {(h,len(discs),min(discs)) for h,discs in small_class_number_discriminants(100).items()}
{(1, 13, -163),
 (2, 29, -427),
 (3, 25, -907),
 (4, 84, -1555),
 (5, 29, -2683),
 (6, 101, -4075),
 (7, 38, -5923),
 (8, 208, -7987),
 (9, 55, -10627),
 (10, 123, -13843),
 (11, 46, -15667),
 (12, 379, -19723),
 (13, 43, -20563),
 (14, 134, -30067),
 (15, 95, -34483),
 (16, 531, -35275),
 (17, 50, -37123),
 (18, 291, -48427),
 (19, 59, -38707),
 (20, 502, -58843),
 (21, 118, -61483),
 (22, 184, -85507),
 (23, 78, -90787),
 (24, 1042, -111763),
 (25, 101, -93307),
 (26, 227, -103027),
 (27, 136, -103387),
 (28, 623, -126043),
 (29, 94, -166147),
 (30, 473, -137083),
 (31, 83, -133387),
 (32, 1231, -164803),
 (33, 158, -222643),
 (34, 261, -189883),
 (35, 111, -210907),
 (36, 1303, -217627),
 (37, 96, -158923),
 (38, 283, -289963),
 (39, 162, -253507),
 (40, 1418, -274003),
 (41, 125, -296587),
 (42, 595, -301387),
 (43, 123, -300787),
 (44, 909, -319867),
 (45, 231, -308323),
 (46, 328, -462883),
 (47, 117, -375523),
 (48, 2893, -335203),
 (49, 146, -393187),
 (50, 443, -389467),
 (51, 217, -546067),
 (52, 1003, -457867),
 (53, 130, -425107),
 (54, 806, -532123),
 (55, 177, -452083),
 (56, 1809, -494323),
 (57, 237, -615883),
 (58, 360, -586987),
 (59, 144, -474307),
 (60, 2352, -662803),
 (61, 149, -606643),
 (62, 386, -647707),
 (63, 311, -991027),
 (64, 2915, -693067),
 (65, 192, -703123),
 (66, 856, -958483),
 (67, 145, -652723),
 (68, 1227, -819163),
 (69, 292, -888427),
 (70, 702, -821683),
 (71, 176, -909547),
 (72, 4046, -947923),
 (73, 137, -886867),
 (74, 472, -951043),
 (75, 353, -916507),
 (76, 1381, -1086187),
 (77, 236, -1242763),
 (78, 921, -1004347),
 (79, 200, -1333963),
 (80, 3851, -1165483),
 (81, 338, -1030723),
 (82, 486, -1446547),
 (83, 174, -1074907),
 (84, 2990, -1225387),
 (85, 246, -1285747),
 (86, 553, -1534723),
 (87, 313, -1261747),
 (88, 2769, -1265587),
 (89, 206, -1429387),
 (90, 1508, -1548523),
 (91, 249, -1391083),
 (92, 1590, -1452067),
 (93, 354, -1475203),
 (94, 598, -1587763),
 (95, 273, -1659067),
 (96, 7265, -1684027),
 (97, 208, -1842523),
 (98, 707, -2383747),
 (99, 396, -1480627),
 (100, 2310, -1856563)}

mdsage.quadratic_class_numbers.small_class_number_fundamental_discriminants(bound=100)¶

returns a dictionary of all imaginary quadratic fundamental discriminants of small class number

INPUT:

bound (optional) - an integer speciefiel

OUTPUT:

a dictionary whose keys are class numbers and whose values are lists of discriminants having that class number

EXAMPLES:

sage: from mdsage import *
sage: small_class_number_fundamental_discriminants(2)
{1: [-3, -4, -7, -8, -11, -19, -43, -67, -163],
 2: [-15,
  -20,
  -24,
  -35,
  -40,
  -51,
  -52,
  -88,
  -91,
  -115,
  -123,
  -148,
  -187,
  -232,
  -235,
  -267,
  -403,
  -427]}
sage: # we verify that our data matched with that of watkins
sage: {(h,len(discs),min(discs)) for h,discs in small_class_number_fundamental_discriminants(100).items()}
{(1, 9, -163),
 (2, 18, -427),
 (3, 16, -907),
 (4, 54, -1555),
 (5, 25, -2683),
 (6, 51, -3763),
 (7, 31, -5923),
 (8, 131, -6307),
 (9, 34, -10627),
 (10, 87, -13843),
 (11, 41, -15667),
 (12, 206, -17803),
 (13, 37, -20563),
 (14, 95, -30067),
 (15, 68, -34483),
 (16, 322, -31243),
 (17, 45, -37123),
 (18, 150, -48427),
 (19, 47, -38707),
 (20, 350, -58507),
 (21, 85, -61483),
 (22, 139, -85507),
 (23, 68, -90787),
 (24, 511, -111763),
 (25, 95, -93307),
 (26, 190, -103027),
 (27, 93, -103387),
 (28, 457, -126043),
 (29, 83, -166147),
 (30, 255, -134467),
 (31, 73, -133387),
 (32, 708, -164803),
 (33, 101, -222643),
 (34, 219, -189883),
 (35, 103, -210907),
 (36, 668, -217627),
 (37, 85, -158923),
 (38, 237, -289963),
 (39, 115, -253507),
 (40, 912, -260947),
 (41, 109, -296587),
 (42, 339, -280267),
 (43, 106, -300787),
 (44, 691, -319867),
 (45, 154, -308323),
 (46, 268, -462883),
 (47, 107, -375523),
 (48, 1365, -335203),
 (49, 132, -393187),
 (50, 345, -389467),
 (51, 159, -546067),
 (52, 770, -439147),
 (53, 114, -425107),
 (54, 427, -532123),
 (55, 163, -452083),
 (56, 1205, -494323),
 (57, 179, -615883),
 (58, 291, -586987),
 (59, 128, -474307),
 (60, 1302, -662803),
 (61, 132, -606643),
 (62, 323, -647707),
 (63, 216, -991027),
 (64, 1672, -693067),
 (65, 164, -703123),
 (66, 530, -958483),
 (67, 120, -652723),
 (68, 976, -819163),
 (69, 209, -888427),
 (70, 560, -811507),
 (71, 150, -909547),
 (72, 1930, -947923),
 (73, 119, -886867),
 (74, 407, -951043),
 (75, 237, -916507),
 (76, 1075, -1086187),
 (77, 216, -1242763),
 (78, 561, -1004347),
 (79, 175, -1333963),
 (80, 2277, -1165483),
 (81, 228, -1030723),
 (82, 402, -1446547),
 (83, 150, -1074907),
 (84, 1715, -1225387),
 (85, 221, -1285747),
 (86, 472, -1534723),
 (87, 222, -1261747),
 (88, 1905, -1265587),
 (89, 192, -1429387),
 (90, 801, -1548523),
 (91, 214, -1391083),
 (92, 1248, -1452067),
 (93, 262, -1475203),
 (94, 509, -1587763),
 (95, 241, -1659067),
 (96, 3283, -1684027),
 (97, 185, -1842523),
 (98, 580, -2383747),
 (99, 289, -1480627),
 (100, 1736, -1856563)}

Class Numbers¶

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