| 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 |
| 19.19.114...721.1 |
$x^{19} - x^{18} - 90 x^{17} + 57 x^{16} + 3044 x^{15} - 1124 x^{14} - 51184 x^{13} + 4822 x^{12} + 474003 x^{11} + 90110 x^{10} - 2465084 x^{9} - 1153239 x^{8} + 6854098 x^{7} + 5023125 x^{6} - 8711114 x^{5} - 8950277 x^{4} + 2600136 x^{3} + 5125792 x^{2} + 1553447 x + 117649$ |
$19$ |
[19,0] |
$191^{18}$ |
$1$ |
$144.870173252$ |
$144.87017325182694$ |
|
✓ |
|
$C_{19}$ (as 19T1) |
trivial |
$2$ |
$18$ |
$2998086774930000$ |
| 19.19.299...561.1 |
$x^{19} - x^{18} - 108 x^{17} + 213 x^{16} + 4270 x^{15} - 11618 x^{14} - 74320 x^{13} + 248057 x^{12} + 559431 x^{11} - 2225275 x^{10} - 1928935 x^{9} + 9223494 x^{8} + 3299920 x^{7} - 17492482 x^{6} - 3737798 x^{5} + 13322064 x^{4} + 4255245 x^{3} - 3318455 x^{2} - 1865379 x - 251347$ |
$19$ |
[19,0] |
$229^{18}$ |
$1$ |
$172.041656242$ |
$172.0416562423144$ |
|
✓ |
|
$C_{19}$ (as 19T1) |
trivial |
$2$ |
$18$ |
$1038259438940000$ |
| 19.19.108...281.1 |
$x^{19} - 171 x^{17} - 133 x^{16} + 11476 x^{15} + 15580 x^{14} - 385833 x^{13} - 673436 x^{12} + 6916190 x^{11} + 13391960 x^{10} - 66283229 x^{9} - 126730380 x^{8} + 339213156 x^{7} + 582575340 x^{6} - 861915924 x^{5} - 1264657480 x^{4} + 868638105 x^{3} + 1138104275 x^{2} - 137550709 x - 221874931$ |
$19$ |
[19,0] |
$19^{36}$ |
$1$ |
$264.789987862$ |
$264.78998786153875$ |
|
✓ |
|
$C_{19}$ (as 19T1) |
trivial |
$2$ |
$18$ |
$50751893334000000$ |
| 19.19.158...641.1 |
$x^{19} - x^{18} - 198 x^{17} + 37 x^{16} + 12055 x^{15} - 1727 x^{14} - 335304 x^{13} + 70692 x^{12} + 4834266 x^{11} - 1201976 x^{10} - 37345852 x^{9} + 7325624 x^{8} + 153105664 x^{7} - 3418584 x^{6} - 328284116 x^{5} - 65770266 x^{4} + 325674937 x^{3} + 129492371 x^{2} - 92941225 x - 41768519$ |
$19$ |
[19,0] |
$419^{18}$ |
$1$ |
$304.931828932$ |
$304.93182893189237$ |
|
✓ |
|
$C_{19}$ (as 19T1) |
trivial |
$2$ |
$18$ |
$1240119351367981000$ |
| 19.19.755...649.1 |
$x^{19} - x^{18} - 216 x^{17} + 329 x^{16} + 17417 x^{15} - 31517 x^{14} - 670894 x^{13} + 1275207 x^{12} + 13251603 x^{11} - 24036830 x^{10} - 137385192 x^{9} + 209490406 x^{8} + 773799359 x^{7} - 862308300 x^{6} - 2305645775 x^{5} + 1693728828 x^{4} + 3278226422 x^{3} - 1551959900 x^{2} - 1701753321 x + 554402371$ |
$19$ |
[19,0] |
$457^{18}$ |
$1$ |
$331.070595959$ |
$331.0705959589611$ |
|
✓ |
|
$C_{19}$ (as 19T1) |
trivial |
$2$ |
$18$ |
$438964507237879700$ |
| 19.19.416...961.1 |
$x^{19} - x^{18} - 270 x^{17} + 351 x^{16} + 28987 x^{15} - 41181 x^{14} - 1648168 x^{13} + 2453428 x^{12} + 55186847 x^{11} - 85340779 x^{10} - 1133336624 x^{9} + 1826548777 x^{8} + 14258200659 x^{7} - 24108274876 x^{6} - 104945874488 x^{5} + 187074906809 x^{4} + 398834944226 x^{3} - 749021546949 x^{2} - 549551190705 x + 1072348621073$ |
$19$ |
[19,0] |
$571^{18}$ |
$1$ |
$408.836815925$ |
$408.8368159245524$ |
|
✓ |
|
$C_{19}$ (as 19T1) |
trivial |
$2$ |
$18$ |
$5743380709748059000$ |
| 19.19.394...889.1 |
$x^{19} - x^{18} - 306 x^{17} - 79 x^{16} + 29370 x^{15} + 130 x^{14} - 1361758 x^{13} + 704483 x^{12} + 34401066 x^{11} - 34840236 x^{10} - 488507308 x^{9} + 678792838 x^{8} + 3902105097 x^{7} - 6509889712 x^{6} - 16762277726 x^{5} + 31790175776 x^{4} + 33454527221 x^{3} - 71781895812 x^{2} - 18736713884 x + 50781641759$ |
$19$ |
[19,0] |
$647^{18}$ |
$1$ |
$460.216250835$ |
$460.2162508352232$ |
|
✓ |
|
$C_{19}$ (as 19T1) |
trivial |
$2$ |
$18$ |
$24719888574728266000$ |
| 19.19.732...481.1 |
$x^{19} - x^{18} - 360 x^{17} - 173 x^{16} + 45376 x^{15} + 58762 x^{14} - 2558302 x^{13} - 4227138 x^{12} + 70534890 x^{11} + 120121397 x^{10} - 973700212 x^{9} - 1501612590 x^{8} + 6678374954 x^{7} + 8595059019 x^{6} - 21259099080 x^{5} - 21796436285 x^{4} + 27241052007 x^{3} + 18814754704 x^{2} - 12659238391 x - 3483379661$ |
$19$ |
[19,0] |
$761^{18}$ |
$1$ |
$536.701529231$ |
$536.7015292310211$ |
|
✓ |
|
$C_{19}$ (as 19T1) |
trivial |
$2$ |
$18$ |
$94833868482425830000$ |
