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.

F_{22}, t^{2}+t+1=0, g=t+1 | |||||
---|---|---|---|---|---|

x | Tr(x) | Generator | Normal | x^{-1} | Tr(x^{-1}) |

{ 1, 2} | 1 | 1 | 1 | { 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 |

F_{23}, t^{3}+t+1=0, g=t^{2} | |||||
---|---|---|---|---|---|

x | Tr(x) | Generator | Normal | x^{-1} | Tr(x^{-1}) |

{ 1, 2, 4} | 0 | 1 | 0 | { 3, 5, 6} | 1 |

{ 3, 5, 6} | 1 | 1 | 1 | { 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 |

F_{24}, t^{4}+t+1=0, g=t | |||||
---|---|---|---|---|---|

x | Tr(x) | Generator | Normal | x^{-1} | Tr(x^{-1}) |

{ 1, 2, 4, 8} | 0 | 1 | 0 | { 7, 11, 13, 14} | 1 |

{ 3, 6, 9, 12} | 1 | 0 | 1 | { 3, 6, 9, 12} | 1 |

{ 7, 11, 13, 14} | 1 | 1 | 1 | { 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 |

F_{25}, t^{5}+t^{2}+1=0, g=t^{2} | |||||
---|---|---|---|---|---|

x | Tr(x) | Generator | Normal | x^{-1} | Tr(x^{-1}) |

{ 1, 2, 4, 8, 16} | 0 | 1 | 0 | { 15, 23, 27, 29, 30} | 0 |

{ 3, 6, 12, 17, 24} | 1 | 1 | 1 | { 7, 14, 19, 25, 28} | 0 |

{ 5, 9, 10, 18, 20} | 1 | 1 | 1 | { 11, 13, 21, 22, 26} | 1 |

{ 7, 14, 19, 25, 28} | 0 | 1 | 0 | { 3, 6, 12, 17, 24} | 1 |

{ 11, 13, 21, 22, 26} | 1 | 1 | 1 | { 5, 9, 10, 18, 20} | 1 |

{ 15, 23, 27, 29, 30} | 0 | 1 | 0 | { 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 |

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 |

F_{26}, t^{6}+t+1=0, g=t^{3}+1 | |||||
---|---|---|---|---|---|

x | Tr(x) | Generator | Normal | x^{-1} | Tr(x^{-1}) |

{ 1, 2, 4, 8, 16, 32} | 0 | 1 | 0 | { 31, 47, 55, 59, 61, 62} | 1 |

{ 3, 6, 12, 24, 33, 48} | 0 | 0 | 0 | { 15, 30, 39, 51, 57, 60} | 1 |

{ 5, 10, 17, 20, 34, 40} | 1 | 1 | 1 | { 23, 29, 43, 46, 53, 58} | 1 |

{ 7, 14, 28, 35, 49, 56} | 0 | 0 | 0 | { 7, 14, 28, 35, 49, 56} | 0 |

{ 11, 22, 25, 37, 44, 50} | 1 | 1 | 0 | { 13, 19, 26, 38, 41, 52} | 0 |

{ 13, 19, 26, 38, 41, 52} | 0 | 1 | 0 | { 11, 22, 25, 37, 44, 50} | 1 |

{ 15, 30, 39, 51, 57, 60} | 1 | 0 | 1 | { 3, 6, 12, 24, 33, 48} | 0 |

{ 23, 29, 43, 46, 53, 58} | 1 | 1 | 1 | { 5, 10, 17, 20, 34, 40} | 1 |

{ 31, 47, 55, 59, 61, 62} | 1 | 1 | 1 | { 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 |

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 |

F_{27}, t^{7}+t+1=0, g=t^{2} | |||||
---|---|---|---|---|---|

x | Tr(x) | Generator | Normal | x^{-1} | Tr(x^{-1}) |

{ 1, 2, 4, 8, 16, 32, 64} | 0 | 1 | 0 | { 63, 95, 111, 119, 123, 125, 126} | 1 |

{ 3, 6, 12, 24, 48, 65, 96} | 0 | 1 | 0 | { 31, 62, 79, 103, 115, 121, 124} | 1 |

{ 5, 10, 20, 33, 40, 66, 80} | 0 | 1 | 0 | { 47, 61, 87, 94, 107, 117, 122} | 1 |

{ 7, 14, 28, 56, 67, 97, 112} | 1 | 1 | 0 | { 15, 30, 60, 71, 99, 113, 120} | 0 |

{ 9, 17, 18, 34, 36, 68, 72} | 0 | 1 | 0 | { 55, 59, 91, 93, 109, 110, 118} | 0 |

{ 11, 22, 44, 49, 69, 88, 98} | 0 | 1 | 0 | { 29, 39, 58, 78, 83, 105, 116} | 0 |

{ 13, 26, 35, 52, 70, 81, 104} | 1 | 1 | 1 | { 23, 46, 57, 75, 92, 101, 114} | 0 |

{ 15, 30, 60, 71, 99, 113, 120} | 0 | 1 | 0 | { 7, 14, 28, 56, 67, 97, 112} | 1 |

{ 19, 25, 38, 50, 73, 76, 100} | 1 | 1 | 1 | { 27, 51, 54, 77, 89, 102, 108} | 1 |

{ 21, 37, 41, 42, 74, 82, 84} | 1 | 1 | 1 | { 43, 45, 53, 85, 86, 90, 106} | 1 |

{ 23, 46, 57, 75, 92, 101, 114} | 0 | 1 | 0 | { 13, 26, 35, 52, 70, 81, 104} | 1 |

{ 27, 51, 54, 77, 89, 102, 108} | 1 | 1 | 1 | { 19, 25, 38, 50, 73, 76, 100} | 1 |

{ 29, 39, 58, 78, 83, 105, 116} | 0 | 1 | 0 | { 11, 22, 44, 49, 69, 88, 98} | 0 |

{ 31, 62, 79, 103, 115, 121, 124} | 1 | 1 | 1 | { 3, 6, 12, 24, 48, 65, 96} | 0 |

{ 43, 45, 53, 85, 86, 90, 106} | 1 | 1 | 1 | { 21, 37, 41, 42, 74, 82, 84} | 1 |

{ 47, 61, 87, 94, 107, 117, 122} | 1 | 1 | 0 | { 5, 10, 20, 33, 40, 66, 80} | 0 |

{ 55, 59, 91, 93, 109, 110, 118} | 0 | 1 | 0 | { 9, 17, 18, 34, 36, 68, 72} | 0 |

{ 63, 95, 111, 119, 123, 125, 126} | 1 | 1 | 1 | { 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 |

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 |

F_{28}, t^{8}+t^{4}+t^{3}+t+1=0, g=t^{2}+1 | |||||
---|---|---|---|---|---|

x | Tr(x) | Generator | Normal | x^{-1} | Tr(x^{-1}) |

{ 1, 2, 4, 8, 16, 32, 64, 128} | 0 | 1 | 0 | {127, 191, 223, 239, 247, 251, 253, 254} | 0 |

{ 3, 6, 12, 24, 48, 96, 129, 192} | 0 | 0 | 0 | { 63, 126, 159, 207, 231, 243, 249, 252} | 1 |

{ 5, 10, 20, 40, 65, 80, 130, 160} | 1 | 0 | 1 | { 95, 125, 175, 190, 215, 235, 245, 250} | 1 |

{ 7, 14, 28, 56, 112, 131, 193, 224} | 0 | 1 | 0 | { 31, 62, 124, 143, 199, 227, 241, 248} | 0 |

{ 9, 18, 33, 36, 66, 72, 132, 144} | 1 | 0 | 1 | {111, 123, 183, 189, 219, 222, 237, 246} | 0 |

{ 11, 22, 44, 88, 97, 133, 176, 194} | 1 | 1 | 1 | { 61, 79, 122, 158, 167, 211, 233, 244} | 1 |

{ 13, 26, 52, 67, 104, 134, 161, 208} | 0 | 1 | 0 | { 47, 94, 121, 151, 188, 203, 229, 242} | 1 |

{ 15, 30, 60, 120, 135, 195, 225, 240} | 1 | 0 | 1 | { 15, 30, 60, 120, 135, 195, 225, 240} | 1 |

{ 19, 38, 49, 76, 98, 137, 152, 196} | 0 | 1 | 0 | { 59, 103, 118, 157, 179, 206, 217, 236} | 0 |

{ 21, 42, 69, 81, 84, 138, 162, 168} | 1 | 0 | 1 | { 87, 93, 117, 171, 174, 186, 213, 234} | 1 |

{ 23, 46, 92, 113, 139, 184, 197, 226} | 0 | 1 | 0 | { 29, 58, 71, 116, 142, 163, 209, 232} | 1 |

{ 25, 35, 50, 70, 100, 140, 145, 200} | 0 | 0 | 0 | { 55, 110, 115, 155, 185, 205, 220, 230} | 1 |

{ 27, 54, 99, 108, 141, 177, 198, 216} | 0 | 0 | 0 | { 39, 57, 78, 114, 147, 156, 201, 228} | 1 |

{ 29, 58, 71, 116, 142, 163, 209, 232} | 1 | 1 | 1 | { 23, 46, 92, 113, 139, 184, 197, 226} | 0 |

{ 31, 62, 124, 143, 199, 227, 241, 248} | 0 | 1 | 0 | { 7, 14, 28, 56, 112, 131, 193, 224} | 0 |

{ 37, 41, 73, 74, 82, 146, 148, 164} | 0 | 1 | 0 | { 91, 107, 109, 173, 181, 182, 214, 218} | 1 |

{ 39, 57, 78, 114, 147, 156, 201, 228} | 1 | 0 | 1 | { 27, 54, 99, 108, 141, 177, 198, 216} | 0 |

{ 43, 86, 89, 101, 149, 172, 178, 202} | 1 | 1 | 1 | { 53, 77, 83, 106, 154, 166, 169, 212} | 1 |

{ 45, 75, 90, 105, 150, 165, 180, 210} | 0 | 0 | 0 | { 45, 75, 90, 105, 150, 165, 180, 210} | 0 |

{ 47, 94, 121, 151, 188, 203, 229, 242} | 1 | 1 | 1 | { 13, 26, 52, 67, 104, 134, 161, 208} | 0 |

{ 53, 77, 83, 106, 154, 166, 169, 212} | 1 | 1 | 1 | { 43, 86, 89, 101, 149, 172, 178, 202} | 1 |

{ 55, 110, 115, 155, 185, 205, 220, 230} | 1 | 0 | 1 | { 25, 35, 50, 70, 100, 140, 145, 200} | 0 |

{ 59, 103, 118, 157, 179, 206, 217, 236} | 0 | 1 | 0 | { 19, 38, 49, 76, 98, 137, 152, 196} | 0 |

{ 61, 79, 122, 158, 167, 211, 233, 244} | 1 | 1 | 1 | { 11, 22, 44, 88, 97, 133, 176, 194} | 1 |

{ 63, 126, 159, 207, 231, 243, 249, 252} | 1 | 0 | 1 | { 3, 6, 12, 24, 48, 96, 129, 192} | 0 |

{ 87, 93, 117, 171, 174, 186, 213, 234} | 1 | 0 | 1 | { 21, 42, 69, 81, 84, 138, 162, 168} | 1 |

{ 91, 107, 109, 173, 181, 182, 214, 218} | 1 | 1 | 1 | { 37, 41, 73, 74, 82, 146, 148, 164} | 0 |

{ 95, 125, 175, 190, 215, 235, 245, 250} | 1 | 0 | 1 | { 5, 10, 20, 40, 65, 80, 130, 160} | 1 |

{111, 123, 183, 189, 219, 222, 237, 246} | 0 | 0 | 0 | { 9, 18, 33, 36, 66, 72, 132, 144} | 1 |

{127, 191, 223, 239, 247, 251, 253, 254} | 0 | 1 | 0 | { 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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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.