| Label |
Polynomial |
Degree |
Signature |
Discriminant |
Ram. prime count |
Root discriminant |
Galois root discriminant |
CM field |
Galois |
Monogenic |
Galois group |
Class group |
Unit group torsion |
Unit group rank |
Regulator |
| 19.1.272...391.1 |
$x^{19} - x^{18} + 2 x^{17} - 5 x^{16} + 8 x^{15} - 14 x^{14} + 13 x^{13} - 10 x^{12} - x^{11} + 9 x^{10} - 18 x^{9} + 25 x^{8} - 10 x^{7} - 4 x^{6} + 38 x^{5} - 42 x^{4} + 37 x^{3} - 16 x^{2} + 4 x + 1$ |
$19$ |
[1,9] |
$-\,311^{9}$ |
$1$ |
$15.1628639032$ |
$17.635192088548397$ |
|
|
? |
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$9$ |
$2166.17030371$ |
| 19.1.990...639.1 |
$x^{19} - 2 x^{18} + 2 x^{17} - 2 x^{16} - 3 x^{15} + 14 x^{14} - 7 x^{13} - 22 x^{12} + 30 x^{11} - 9 x^{10} + 5 x^{9} - 2 x^{8} - 51 x^{7} + 90 x^{6} - 19 x^{5} - 91 x^{4} + 113 x^{3} - 59 x^{2} + 14 x - 1$ |
$19$ |
[1,9] |
$-\,359^{9}$ |
$1$ |
$16.2296030083$ |
$18.947295321496416$ |
|
|
? |
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$9$ |
$4771.45184867$ |
| 19.1.467...879.1 |
$x^{19} - 5 x^{18} + 10 x^{17} - 5 x^{16} + 9 x^{15} - 42 x^{14} + 51 x^{13} - 6 x^{12} - 23 x^{11} + 4 x^{10} - 29 x^{9} + 112 x^{8} - 63 x^{7} - 147 x^{6} + 183 x^{5} + 15 x^{4} - 77 x^{3} - 17 x^{2} + 37 x + 1$ |
$19$ |
[1,9] |
$-\,919^{9}$ |
$1$ |
$25.3323668457$ |
$30.315012782448235$ |
|
|
|
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$9$ |
$1152808.1284$ |
| 19.1.173...823.1 |
$x^{19} + 9 x^{17} - 10 x^{16} + 8 x^{15} - 45 x^{14} + 39 x^{13} + 7 x^{12} + 42 x^{11} - 36 x^{10} - 60 x^{9} + 9 x^{8} + 55 x^{7} + 63 x^{6} + 15 x^{5} - 88 x^{4} - 7 x^{3} + 27 x^{2} + 51 x + 1$ |
$19$ |
[1,9] |
$-\,1063^{9}$ |
$1$ |
$27.140698466$ |
$32.60368077380221$ |
|
|
|
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$9$ |
$2724154.30052$ |
| 19.1.512...799.1 |
$x^{19} - 3 x^{18} + 17 x^{17} - 47 x^{16} + 135 x^{15} - 310 x^{14} + 709 x^{13} - 1319 x^{12} + 2423 x^{11} - 3757 x^{10} + 5497 x^{9} - 6561 x^{8} + 7142 x^{7} - 5699 x^{6} + 4212 x^{5} - 1490 x^{4} + 482 x^{3} + 408 x^{2} - 210 x + 49$ |
$19$ |
[1,9] |
$-\,11^{9}\cdot 109^{9}$ |
$2$ |
$28.7334671769$ |
$34.62657938636157$ |
|
|
? |
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$9$ |
$2481320.5796$ |
| 19.1.473...375.1 |
$x^{19} - 6 x^{18} + 18 x^{17} - 22 x^{16} - 38 x^{15} + 219 x^{14} - 403 x^{13} + 380 x^{12} - 536 x^{11} + 1378 x^{10} - 1608 x^{9} + 691 x^{8} - 1073 x^{7} + 1881 x^{6} - 757 x^{5} + 716 x^{4} - 1464 x^{3} + 513 x^{2} - 10 x + 295$ |
$19$ |
[1,9] |
$-\,5^{9}\cdot 307^{9}$ |
$2$ |
$32.3004990658$ |
$39.179076048319466$ |
|
|
|
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$9$ |
$26027741.7153$ |
| 19.1.495...343.1 |
$x^{19} - 9 x^{18} + 24 x^{17} - 6 x^{16} - 42 x^{15} + 4 x^{14} + 34 x^{13} + 74 x^{12} + 46 x^{11} - 238 x^{10} - 49 x^{9} + 181 x^{8} + 195 x^{7} + 116 x^{6} - 193 x^{5} - 275 x^{4} + 106 x^{3} - 8 x^{2} + 121 x - 1$ |
$19$ |
[1,9] |
$-\,1543^{9}$ |
$1$ |
$32.3801306357$ |
$39.28103868280471$ |
|
|
|
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$9$ |
$17317158.8896$ |
| 19.1.101...431.1 |
$x^{19} - 5 x^{18} + 11 x^{17} - 10 x^{16} + 3 x^{15} + 3 x^{14} - 80 x^{13} + 481 x^{12} - 1180 x^{11} + 1220 x^{10} + 888 x^{9} - 5070 x^{8} + 8854 x^{7} - 9440 x^{6} + 6893 x^{5} - 3748 x^{4} + 2592 x^{3} - 2790 x^{2} + 2673 x - 1053$ |
$19$ |
[1,9] |
$-\,3^{9}\cdot 557^{9}$ |
$2$ |
$33.6258332676$ |
$40.87786687193939$ |
|
|
|
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$9$ |
$191261859.157$ |
| 19.1.231...871.1 |
$x^{19} - 6 x^{18} + 18 x^{17} - 39 x^{16} + 73 x^{15} - 200 x^{14} + 265 x^{13} + 305 x^{12} - 931 x^{11} + 1905 x^{10} - 5214 x^{9} + 10284 x^{8} - 13343 x^{7} + 12719 x^{6} - 8662 x^{5} + 4443 x^{4} - 1732 x^{3} + 614 x^{2} - 152 x + 39$ |
$19$ |
[1,9] |
$-\,1831^{9}$ |
$1$ |
$35.1142949338$ |
$42.790185790669334$ |
|
|
|
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$9$ |
$34462435.5384$ |
| 19.1.336...139.1 |
$x^{19} - 38 x^{17} + 703 x^{15} - 228 x^{14} - 7239 x^{13} + 6384 x^{12} + 39159 x^{11} - 64068 x^{10} - 78147 x^{9} + 263112 x^{8} - 144818 x^{7} - 236892 x^{6} + 532855 x^{5} - 695400 x^{4} + 696844 x^{3} - 445968 x^{2} + 229824 x - 110592$ |
$19$ |
[1,9] |
$-\,19^{27}$ |
$1$ |
$65.641148293$ |
$70.9296113719033$ |
|
|
|
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$9$ |
$215015238772$ |
| 19.19.222...669.1 |
$x^{19} - 2 x^{18} - 78 x^{17} + 264 x^{16} + 1763 x^{15} - 8614 x^{14} - 9209 x^{13} + 96924 x^{12} - 60856 x^{11} - 443620 x^{10} + 727151 x^{9} + 587984 x^{8} - 2232580 x^{7} + 1188184 x^{6} + 1637520 x^{5} - 2688288 x^{4} + 1618560 x^{3} - 475264 x^{2} + 61952 x - 2048$ |
$19$ |
[19,0] |
$18229^{9}$ |
$1$ |
$104.293221959$ |
$135.0148140020198$ |
|
|
|
$D_{19}$ (as 19T2) |
trivial |
$2$ |
$18$ |
$253663561168000$ |
