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Brute force counting data for b = 1

The page contains brute force results of testsuite/point/bspacetables3.cc for various countable characteristics that turned out not to be depending on the chosen generator.

m = 2

F22, t2+t+1=0, g=t+1
xTr(x)GeneratorNormalx-1Tr(x-1)
{ 1, 2}111{ 1, 2}1
Number of Frobenius equivalence classes: 1
Number of Frobenius equivalence classes with trace 1: 1
Number of Frobenius equivalence classes that are their own inverse: 1
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 1
Number of generators: 1
Number of generators with trace 1 (no formula): 1
Number of normal basis: 1
Number of normal basis that exist of generators (no formula, A107222): 1
Number of solutions, w(2, 1): 1
Number of solutions with trace 1 (no formula): 1
Number of solutions that are generators (no formula): 1
Number of solutions that are generators with trace 1 (no formula): 1
Number of solutions that are normal basis (no formula): 1
Number of solutions that are normal basis and generators (no formula): 1

m = 3

F23, t3+t+1=0, g=t2
xTr(x)GeneratorNormalx-1Tr(x-1)
{ 1, 2, 4}010{ 3, 5, 6}1
{ 3, 5, 6}111{ 1, 2, 4}0
Number of Frobenius equivalence classes: 2
Number of Frobenius equivalence classes with trace 1: 1
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 2
Number of generators with trace 1 (no formula): 1
Number of normal basis: 1
Number of normal basis that exist of generators (no formula, A107222): 1
Number of solutions, w(3, 1): 0
Number of solutions with trace 1 (no formula): 0
Number of solutions that are generators (no formula): 0
Number of solutions that are generators with trace 1 (no formula): 0
Number of solutions that are normal basis (no formula): 0
Number of solutions that are normal basis and generators (no formula): 0

m = 4

F24, t4+t+1=0, g=t
xTr(x)GeneratorNormalx-1Tr(x-1)
{ 1, 2, 4, 8}010{ 7, 11, 13, 14}1
{ 3, 6, 9, 12}101{ 3, 6, 9, 12}1
{ 7, 11, 13, 14}111{ 1, 2, 4, 8}0
Number of Frobenius equivalence classes: 3
Number of Frobenius equivalence classes with trace 1: 2
Number of Frobenius equivalence classes that are their own inverse: 1
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 1
Number of generators: 2
Number of generators with trace 1 (no formula): 1
Number of normal basis: 2
Number of normal basis that exist of generators (no formula, A107222): 1
Number of solutions, w(4, 1): 1
Number of solutions with trace 1 (no formula): 1
Number of solutions that are generators (no formula): 0
Number of solutions that are generators with trace 1 (no formula): 0
Number of solutions that are normal basis (no formula): 1
Number of solutions that are normal basis and generators (no formula): 0

m = 5

F25, t5+t2+1=0, g=t2
xTr(x)GeneratorNormalx-1Tr(x-1)
{ 1, 2, 4, 8, 16}010{ 15, 23, 27, 29, 30}0
{ 3, 6, 12, 17, 24}111{ 7, 14, 19, 25, 28}0
{ 5, 9, 10, 18, 20}111{ 11, 13, 21, 22, 26}1
{ 7, 14, 19, 25, 28}010{ 3, 6, 12, 17, 24}1
{ 11, 13, 21, 22, 26}111{ 5, 9, 10, 18, 20}1
{ 15, 23, 27, 29, 30}010{ 1, 2, 4, 8, 16}0
Number of Frobenius equivalence classes: 6
Number of Frobenius equivalence classes with trace 1: 3
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 6
Number of generators with trace 1 (no formula): 3
Number of normal basis: 3
Number of normal basis that exist of generators (no formula, A107222): 3
Number of solutions, w(5, 1): 4
Number of solutions with trace 1 (no formula): 2
Number of solutions that are generators (no formula): 4
Number of solutions that are generators with trace 1 (no formula): 2
Number of solutions that are normal basis (no formula): 2
Number of solutions that are normal basis and generators (no formula): 2

m = 6

F26, t6+t+1=0, g=t3+1
xTr(x)GeneratorNormalx-1Tr(x-1)
{ 1, 2, 4, 8, 16, 32}010{ 31, 47, 55, 59, 61, 62}1
{ 3, 6, 12, 24, 33, 48}000{ 15, 30, 39, 51, 57, 60}1
{ 5, 10, 17, 20, 34, 40}111{ 23, 29, 43, 46, 53, 58}1
{ 7, 14, 28, 35, 49, 56}000{ 7, 14, 28, 35, 49, 56}0
{ 11, 22, 25, 37, 44, 50}110{ 13, 19, 26, 38, 41, 52}0
{ 13, 19, 26, 38, 41, 52}010{ 11, 22, 25, 37, 44, 50}1
{ 15, 30, 39, 51, 57, 60}101{ 3, 6, 12, 24, 33, 48}0
{ 23, 29, 43, 46, 53, 58}111{ 5, 10, 17, 20, 34, 40}1
{ 31, 47, 55, 59, 61, 62}111{ 1, 2, 4, 8, 16, 32}0
Number of Frobenius equivalence classes: 9
Number of Frobenius equivalence classes with trace 1: 5
Number of Frobenius equivalence classes that are their own inverse: 1
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 6
Number of generators with trace 1 (no formula): 4
Number of normal basis: 4
Number of normal basis that exist of generators (no formula, A107222): 3
Number of solutions, w(6, 1): 3
Number of solutions with trace 1 (no formula): 2
Number of solutions that are generators (no formula): 2
Number of solutions that are generators with trace 1 (no formula): 2
Number of solutions that are normal basis (no formula): 2
Number of solutions that are normal basis and generators (no formula): 2

m = 7

F27, t7+t+1=0, g=t2
xTr(x)GeneratorNormalx-1Tr(x-1)
{ 1, 2, 4, 8, 16, 32, 64}010{ 63, 95, 111, 119, 123, 125, 126}1
{ 3, 6, 12, 24, 48, 65, 96}010{ 31, 62, 79, 103, 115, 121, 124}1
{ 5, 10, 20, 33, 40, 66, 80}010{ 47, 61, 87, 94, 107, 117, 122}1
{ 7, 14, 28, 56, 67, 97, 112}110{ 15, 30, 60, 71, 99, 113, 120}0
{ 9, 17, 18, 34, 36, 68, 72}010{ 55, 59, 91, 93, 109, 110, 118}0
{ 11, 22, 44, 49, 69, 88, 98}010{ 29, 39, 58, 78, 83, 105, 116}0
{ 13, 26, 35, 52, 70, 81, 104}111{ 23, 46, 57, 75, 92, 101, 114}0
{ 15, 30, 60, 71, 99, 113, 120}010{ 7, 14, 28, 56, 67, 97, 112}1
{ 19, 25, 38, 50, 73, 76, 100}111{ 27, 51, 54, 77, 89, 102, 108}1
{ 21, 37, 41, 42, 74, 82, 84}111{ 43, 45, 53, 85, 86, 90, 106}1
{ 23, 46, 57, 75, 92, 101, 114}010{ 13, 26, 35, 52, 70, 81, 104}1
{ 27, 51, 54, 77, 89, 102, 108}111{ 19, 25, 38, 50, 73, 76, 100}1
{ 29, 39, 58, 78, 83, 105, 116}010{ 11, 22, 44, 49, 69, 88, 98}0
{ 31, 62, 79, 103, 115, 121, 124}111{ 3, 6, 12, 24, 48, 65, 96}0
{ 43, 45, 53, 85, 86, 90, 106}111{ 21, 37, 41, 42, 74, 82, 84}1
{ 47, 61, 87, 94, 107, 117, 122}110{ 5, 10, 20, 33, 40, 66, 80}0
{ 55, 59, 91, 93, 109, 110, 118}010{ 9, 17, 18, 34, 36, 68, 72}0
{ 63, 95, 111, 119, 123, 125, 126}111{ 1, 2, 4, 8, 16, 32, 64}0
Number of Frobenius equivalence classes: 18
Number of Frobenius equivalence classes with trace 1: 9
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 18
Number of generators with trace 1 (no formula): 9
Number of normal basis: 7
Number of normal basis that exist of generators (no formula, A107222): 7
Number of solutions, w(7, 1): 8
Number of solutions with trace 1 (no formula): 4
Number of solutions that are generators (no formula): 8
Number of solutions that are generators with trace 1 (no formula): 4
Number of solutions that are normal basis (no formula): 4
Number of solutions that are normal basis and generators (no formula): 4

m = 8

F28, t8+t4+t3+t+1=0, g=t2+1
xTr(x)GeneratorNormalx-1Tr(x-1)
{ 1, 2, 4, 8, 16, 32, 64, 128}010{127, 191, 223, 239, 247, 251, 253, 254}0
{ 3, 6, 12, 24, 48, 96, 129, 192}000{ 63, 126, 159, 207, 231, 243, 249, 252}1
{ 5, 10, 20, 40, 65, 80, 130, 160}101{ 95, 125, 175, 190, 215, 235, 245, 250}1
{ 7, 14, 28, 56, 112, 131, 193, 224}010{ 31, 62, 124, 143, 199, 227, 241, 248}0
{ 9, 18, 33, 36, 66, 72, 132, 144}101{111, 123, 183, 189, 219, 222, 237, 246}0
{ 11, 22, 44, 88, 97, 133, 176, 194}111{ 61, 79, 122, 158, 167, 211, 233, 244}1
{ 13, 26, 52, 67, 104, 134, 161, 208}010{ 47, 94, 121, 151, 188, 203, 229, 242}1
{ 15, 30, 60, 120, 135, 195, 225, 240}101{ 15, 30, 60, 120, 135, 195, 225, 240}1
{ 19, 38, 49, 76, 98, 137, 152, 196}010{ 59, 103, 118, 157, 179, 206, 217, 236}0
{ 21, 42, 69, 81, 84, 138, 162, 168}101{ 87, 93, 117, 171, 174, 186, 213, 234}1
{ 23, 46, 92, 113, 139, 184, 197, 226}010{ 29, 58, 71, 116, 142, 163, 209, 232}1
{ 25, 35, 50, 70, 100, 140, 145, 200}000{ 55, 110, 115, 155, 185, 205, 220, 230}1
{ 27, 54, 99, 108, 141, 177, 198, 216}000{ 39, 57, 78, 114, 147, 156, 201, 228}1
{ 29, 58, 71, 116, 142, 163, 209, 232}111{ 23, 46, 92, 113, 139, 184, 197, 226}0
{ 31, 62, 124, 143, 199, 227, 241, 248}010{ 7, 14, 28, 56, 112, 131, 193, 224}0
{ 37, 41, 73, 74, 82, 146, 148, 164}010{ 91, 107, 109, 173, 181, 182, 214, 218}1
{ 39, 57, 78, 114, 147, 156, 201, 228}101{ 27, 54, 99, 108, 141, 177, 198, 216}0
{ 43, 86, 89, 101, 149, 172, 178, 202}111{ 53, 77, 83, 106, 154, 166, 169, 212}1
{ 45, 75, 90, 105, 150, 165, 180, 210}000{ 45, 75, 90, 105, 150, 165, 180, 210}0
{ 47, 94, 121, 151, 188, 203, 229, 242}111{ 13, 26, 52, 67, 104, 134, 161, 208}0
{ 53, 77, 83, 106, 154, 166, 169, 212}111{ 43, 86, 89, 101, 149, 172, 178, 202}1
{ 55, 110, 115, 155, 185, 205, 220, 230}101{ 25, 35, 50, 70, 100, 140, 145, 200}0
{ 59, 103, 118, 157, 179, 206, 217, 236}010{ 19, 38, 49, 76, 98, 137, 152, 196}0
{ 61, 79, 122, 158, 167, 211, 233, 244}111{ 11, 22, 44, 88, 97, 133, 176, 194}1
{ 63, 126, 159, 207, 231, 243, 249, 252}101{ 3, 6, 12, 24, 48, 96, 129, 192}0
{ 87, 93, 117, 171, 174, 186, 213, 234}101{ 21, 42, 69, 81, 84, 138, 162, 168}1
{ 91, 107, 109, 173, 181, 182, 214, 218}111{ 37, 41, 73, 74, 82, 146, 148, 164}0
{ 95, 125, 175, 190, 215, 235, 245, 250}101{ 5, 10, 20, 40, 65, 80, 130, 160}1
{111, 123, 183, 189, 219, 222, 237, 246}000{ 9, 18, 33, 36, 66, 72, 132, 144}1
{127, 191, 223, 239, 247, 251, 253, 254}010{ 1, 2, 4, 8, 16, 32, 64, 128}0
Number of Frobenius equivalence classes: 30
Number of Frobenius equivalence classes with trace 1: 16
Number of Frobenius equivalence classes that are their own inverse: 2
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 1
Number of generators: 16
Number of generators with trace 1 (no formula): 7
Number of normal basis: 16
Number of normal basis that exist of generators (no formula, A107222): 7
Number of solutions, w(8, 1): 16
Number of solutions with trace 1 (no formula): 9
Number of solutions that are generators (no formula): 10
Number of solutions that are generators with trace 1 (no formula): 4
Number of solutions that are normal basis (no formula): 9
Number of solutions that are normal basis and generators (no formula): 4

m = 9

Number of Frobenius equivalence classes: 56
Number of Frobenius equivalence classes with trace 1: 28
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 48
Number of generators with trace 1 (no formula): 25
Number of normal basis: 21
Number of normal basis that exist of generators (no formula, A107222): 19
Number of solutions, w(9, 1): 28
Number of solutions with trace 1 (no formula): 14
Number of solutions that are generators (no formula): 26
Number of solutions that are generators with trace 1 (no formula): 14
Number of solutions that are normal basis (no formula): 12
Number of solutions that are normal basis and generators (no formula): 12

m = 10

Number of Frobenius equivalence classes: 99
Number of Frobenius equivalence classes with trace 1: 51
Number of Frobenius equivalence classes that are their own inverse: 3
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 2
Number of generators: 60
Number of generators with trace 1 (no formula): 31
Number of normal basis: 48
Number of normal basis that exist of generators (no formula, A107222): 29
Number of solutions, w(10, 1): 45
Number of solutions with trace 1 (no formula): 24
Number of solutions that are generators (no formula): 26
Number of solutions that are generators with trace 1 (no formula): 14
Number of solutions that are normal basis (no formula): 22
Number of solutions that are normal basis and generators (no formula): 13

m = 11

Number of Frobenius equivalence classes: 186
Number of Frobenius equivalence classes with trace 1: 93
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 176
Number of generators with trace 1 (no formula): 87
Number of normal basis: 93
Number of normal basis that exist of generators (no formula, A107222): 87
Number of solutions, w(11, 1): 96
Number of solutions with trace 1 (no formula): 48
Number of solutions that are generators (no formula): 94
Number of solutions that are generators with trace 1 (no formula): 46
Number of solutions that are normal basis (no formula): 48
Number of solutions that are normal basis and generators (no formula): 46

m = 12

Number of Frobenius equivalence classes: 335
Number of Frobenius equivalence classes with trace 1: 170
Number of Frobenius equivalence classes that are their own inverse: 5
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 2
Number of generators: 144
Number of generators with trace 1 (no formula): 72
Number of normal basis: 128
Number of normal basis that exist of generators (no formula, A107222): 52
Number of solutions, w(12, 1): 167
Number of solutions with trace 1 (no formula): 86
Number of solutions that are generators (no formula): 76
Number of solutions that are generators with trace 1 (no formula): 38
Number of solutions that are normal basis (no formula): 62
Number of solutions that are normal basis and generators (no formula): 25

m = 13

Number of Frobenius equivalence classes: 630
Number of Frobenius equivalence classes with trace 1: 315
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 630
Number of generators with trace 1 (no formula): 315
Number of normal basis: 315
Number of normal basis that exist of generators (no formula, A107222): 315
Number of solutions, w(13, 1): 308
Number of solutions with trace 1 (no formula): 154
Number of solutions that are generators (no formula): 308
Number of solutions that are generators with trace 1 (no formula): 154
Number of solutions that are normal basis (no formula): 154
Number of solutions that are normal basis and generators (no formula): 154

m = 14

Number of Frobenius equivalence classes: 1161
Number of Frobenius equivalence classes with trace 1: 585
Number of Frobenius equivalence classes that are their own inverse: 9
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 4
Number of generators: 756
Number of generators with trace 1 (no formula): 381
Number of normal basis: 448
Number of normal basis that exist of generators (no formula, A107222): 291
Number of solutions, w(14, 1): 579
Number of solutions with trace 1 (no formula): 294
Number of solutions that are generators (no formula): 378
Number of solutions that are generators with trace 1 (no formula): 192
Number of solutions that are normal basis (no formula): 224
Number of solutions that are normal basis and generators (no formula): 145

m = 15

Number of Frobenius equivalence classes: 2182
Number of Frobenius equivalence classes with trace 1: 1091
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 1800
Number of generators with trace 1 (no formula): 901
Number of normal basis: 675
Number of normal basis that exist of generators (no formula, A107222): 562
Number of solutions, w(15, 1): 1100
Number of solutions with trace 1 (no formula): 550
Number of solutions that are generators (no formula): 902
Number of solutions that are generators with trace 1 (no formula): 452
Number of solutions that are normal basis (no formula): 332
Number of solutions that are normal basis and generators (no formula): 277

m = 16

Number of Frobenius equivalence classes: 4080
Number of Frobenius equivalence classes with trace 1: 2048
Number of Frobenius equivalence classes that are their own inverse: 16
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 9
Number of generators: 2048
Number of generators with trace 1 (no formula): 1017
Number of normal basis: 2048
Number of normal basis that exist of generators (no formula, A107222): 1017
Number of solutions, w(16, 1): 2018
Number of solutions with trace 1 (no formula): 1017
Number of solutions that are generators (no formula): 1014
Number of solutions that are generators with trace 1 (no formula): 500
Number of solutions that are normal basis (no formula): 1017
Number of solutions that are normal basis and generators (no formula): 500

m = 17

Number of Frobenius equivalence classes: 7710
Number of Frobenius equivalence classes with trace 1: 3855
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 7710
Number of generators with trace 1 (no formula): 3855
Number of normal basis: 3825
Number of normal basis that exist of generators (no formula, A107222): 3825
Number of solutions, w(17, 1): 3852
Number of solutions with trace 1 (no formula): 1926
Number of solutions that are generators (no formula): 3852
Number of solutions that are generators with trace 1 (no formula): 1926
Number of solutions that are normal basis (no formula): 1912
Number of solutions that are normal basis and generators (no formula): 1912

m = 18

Number of Frobenius equivalence classes: 14532
Number of Frobenius equivalence classes with trace 1: 7280
Number of Frobenius equivalence classes that are their own inverse: 28
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 14
Number of generators: 7776
Number of generators with trace 1 (no formula): 3890
Number of normal basis: 5376
Number of normal basis that exist of generators (no formula, A107222): 2870
Number of solutions, w(18, 1): 7280
Number of solutions with trace 1 (no formula): 3654
Number of solutions that are generators (no formula): 3908
Number of solutions that are generators with trace 1 (no formula): 1956
Number of solutions that are normal basis (no formula): 2712
Number of solutions that are normal basis and generators (no formula): 1459

m = 19

Number of Frobenius equivalence classes: 27594
Number of Frobenius equivalence classes with trace 1: 13797
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 27594
Number of generators with trace 1 (no formula): 13797
Number of normal basis: 13797
Number of normal basis that exist of generators (no formula, A107222): 13797
Number of solutions, w(19, 1): 13776
Number of solutions with trace 1 (no formula): 6888
Number of solutions that are generators (no formula): 13776
Number of solutions that are generators with trace 1 (no formula): 6888
Number of solutions that are normal basis (no formula): 6888
Number of solutions that are normal basis and generators (no formula): 6888

m = 20

Number of Frobenius equivalence classes: 52377
Number of Frobenius equivalence classes with trace 1: 26214
Number of Frobenius equivalence classes that are their own inverse: 51
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 24
Number of generators: 24000
Number of generators with trace 1 (no formula): 12000
Number of normal basis: 24576
Number of normal basis that exist of generators (no formula, A107222): 11255
Number of solutions, w(20, 1): 26133
Number of solutions with trace 1 (no formula): 13092
Number of solutions that are generators (no formula): 11944
Number of solutions that are generators with trace 1 (no formula): 5972
Number of solutions that are normal basis (no formula): 12286
Number of solutions that are normal basis and generators (no formula): 5603

m = 21

Number of Frobenius equivalence classes: 99858
Number of Frobenius equivalence classes with trace 1: 49929
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 84672
Number of generators with trace 1 (no formula): 42344
Number of normal basis: 27783
Number of normal basis that exist of generators (no formula, A107222): 23579
Number of solutions, w(21, 1): 49996
Number of solutions with trace 1 (no formula): 24998
Number of solutions that are generators (no formula): 42460
Number of solutions that are generators with trace 1 (no formula): 21238
Number of solutions that are normal basis (no formula): 13912
Number of solutions that are normal basis and generators (no formula): 11876

m = 22

Number of Frobenius equivalence classes: 190557
Number of Frobenius equivalence classes with trace 1: 95325
Number of Frobenius equivalence classes that are their own inverse: 93
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 48
Number of generators: 120032
Number of generators with trace 1 (no formula): 60043
Number of normal basis: 95232
Number of normal basis that exist of generators (no formula, A107222): 59986
Number of solutions, w(22, 1): 95223
Number of solutions with trace 1 (no formula): 47658
Number of solutions that are generators (no formula): 60014
Number of solutions that are generators with trace 1 (no formula): 30034
Number of solutions that are normal basis (no formula): 47610
Number of solutions that are normal basis and generators (no formula): 30005

m = 23

Number of Frobenius equivalence classes: 364722
Number of Frobenius equivalence classes with trace 1: 182361
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 356960
Number of generators with trace 1 (no formula): 178431
Number of normal basis: 182183
Number of normal basis that exist of generators (no formula, A107222): 178259
Number of solutions, w(23, 1): 182248
Number of solutions with trace 1 (no formula): 91124
Number of solutions that are generators (no formula): 178414
Number of solutions that are generators with trace 1 (no formula): 89158
Number of solutions that are normal basis (no formula): 91028
Number of solutions that are normal basis and generators (no formula): 89062

m = 24

Number of Frobenius equivalence classes: 698870
Number of Frobenius equivalence classes with trace 1: 349520
Number of Frobenius equivalence classes that are their own inverse: 170
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 86
Number of generators: 276480
Number of generators with trace 1 (no formula): 138224
Number of normal basis: 262144
Number of normal basis that exist of generators (no formula, A107222): 103680
Number of solutions, w(24, 1): 349474
Number of solutions with trace 1 (no formula): 174822
Number of solutions that are generators (no formula): 138360
Number of solutions that are generators with trace 1 (no formula): 69164
Number of solutions that are normal basis (no formula): 131094
Number of solutions that are normal basis and generators (no formula): 51888

m = 25

Number of Frobenius equivalence classes: 1342176
Number of Frobenius equivalence classes with trace 1: 671088
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 1296000
Number of generators with trace 1 (no formula): 648031
Number of normal basis: 629145
Number of normal basis that exist of generators (no formula, A107222): 607522
Number of solutions, w(25, 1): 671176
Number of solutions with trace 1 (no formula): 335588
Number of solutions that are generators (no formula): 648042
Number of solutions that are generators with trace 1 (no formula): 324052
Number of solutions that are normal basis (no formula): 314584
Number of solutions that are normal basis and generators (no formula): 303761

m = 26

Number of Frobenius equivalence classes: 2580795
Number of Frobenius equivalence classes with trace 1: 1290555
Number of Frobenius equivalence classes that are their own inverse: 315
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 154
Number of generators: 1719900
Number of generators with trace 1 (no formula): 860059
Number of normal basis: 1290240
Number of normal basis that exist of generators (no formula, A107222): 859849
Number of solutions, w(26, 1): 1289925
Number of solutions with trace 1 (no formula): 645120
Number of solutions that are generators (no formula): 859642
Number of solutions that are generators with trace 1 (no formula): 429930
Number of solutions that are normal basis (no formula): 644966
Number of solutions that are normal basis and generators (no formula): 429829

m = 27

Number of Frobenius equivalence classes: 4971008
Number of Frobenius equivalence classes with trace 1: 2485504
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 4202496
Number of generators with trace 1 (no formula): 2101353
Number of normal basis: 1835001
Number of normal basis that exist of generators (no formula, A107222): 1551227
Number of solutions, w(27, 1): 2485644
Number of solutions with trace 1 (no formula): 1242822
Number of solutions that are generators (no formula): 2101290
Number of solutions that are generators with trace 1 (no formula): 1050750
Number of solutions that are normal basis (no formula): 917600
Number of solutions that are normal basis and generators (no formula): 775566

m = 28

Number of Frobenius equivalence classes: 9586395
Number of Frobenius equivalence classes with trace 1: 4793490
Number of Frobenius equivalence classes that are their own inverse: 585
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 294
Number of generators: 4741632
Number of generators with trace 1 (no formula): 2370715
Number of normal basis: 3670016
Number of normal basis that exist of generators (no formula, A107222): 1815045
Number of solutions, w(28, 1): 4793355
Number of solutions with trace 1 (no formula): 2396970
Number of solutions that are generators (no formula): 2370858
Number of solutions that are generators with trace 1 (no formula): 1185328
Number of solutions that are normal basis (no formula): 1835144
Number of solutions that are normal basis and generators (no formula): 907684

m = 29

Number of Frobenius equivalence classes: 18512790
Number of Frobenius equivalence classes with trace 1: 9256395
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 18407808
Number of generators with trace 1 (no formula): 9203747
Number of normal basis: 9256395
Number of normal basis that exist of generators (no formula, A107222): 9203747
Number of solutions, w(29, 1): 9255700
Number of solutions with trace 1 (no formula): 4627850
Number of solutions that are generators (no formula): 9202954
Number of solutions that are generators with trace 1 (no formula): 4601320
Number of solutions that are normal basis (no formula): 4627850
Number of solutions that are normal basis and generators (no formula): 4601320

m = 30

Number of Frobenius equivalence classes: 35790267
Number of Frobenius equivalence classes with trace 1: 17895679
Number of Frobenius equivalence classes that are their own inverse: 1091
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 550
Number of generators: 17820000
Number of generators with trace 1 (no formula): 8908940
Number of normal basis: 11059200
Number of normal basis that exist of generators (no formula, A107222): 5505966
Number of solutions, w(30, 1): 17894421
Number of solutions with trace 1 (no formula): 8947756
Number of solutions that are generators (no formula): 8910288
Number of solutions that are generators with trace 1 (no formula): 4454084
Number of solutions that are normal basis (no formula): 5529896
Number of solutions that are normal basis and generators (no formula): 2753361

m = 31

Number of Frobenius equivalence classes: 69273666
Number of Frobenius equivalence classes with trace 1: 34636833
Number of Frobenius equivalence classes that are their own inverse: 0
Number of Frobenius equivalence classes that are their own inverse with trace 1 (depends on w(m/2, 1)): 0
Number of generators: 69273666
Number of generators with trace 1 (no formula): 34636833
Number of normal basis: 28629151
Number of normal basis that exist of generators (no formula, A107222): 28629151
Number of solutions, w(31, 1): 34638296
Number of solutions with trace 1 (no formula): 17319148
Number of solutions that are generators (no formula): 34638296
Number of solutions that are generators with trace 1 (no formula): 17319148
Number of solutions that are normal basis (no formula): 14314976
Number of solutions that are normal basis and generators (no formula): 14314976

A summary can be found on Cracking parameter b of the elliptic curve.

Copyright © 2002-2008 Carlo Wood.  All rights reserved.