| Label |
Polynomial |
Degree |
Signature |
Discriminant |
Ram. prime count |
Root discriminant |
Galois root discriminant |
CM field |
Galois |
Monogenic |
Galois group |
Class group |
Narrow class group |
Unit group torsion |
Unit group rank |
Regulator |
Unit signature rank |
Max $p$ |
| 18.0.270...307.1 |
$x^{18} + 3 x^{16} - 276 x^{15} - 405 x^{14} - 1101 x^{13} - 5106 x^{12} + 72054 x^{11} + 252957 x^{10} + 2968148 x^{9} + 3249099 x^{8} - 10766211 x^{7} + 211383570 x^{6} - 235676520 x^{5} + 114042921 x^{4} + 1994966916 x^{3} - 1233883107 x^{2} - 2414714757 x + 9447645601$ |
$18$ |
(0, 9) |
$-\,3^{31}\cdot 137^{12}$ |
$2$ |
$176.274383996$ |
$176.27438399591628$ |
|
✓ |
? |
$S_3 \times C_3$ (as 18T3) |
$[2, 2, 14, 14]$ |
$[2, 2, 14, 14]$ |
$18$ |
$8$ |
$113730305559.83772$ |
$0$ |
$137$ |
| 18.0.533...000.2 |
$x^{18} - 8 x^{15} + 35344 x^{12} + 529920 x^{9} + 381715200 x^{6} - 933120000 x^{3} + 1259712000000$ |
$18$ |
(0, 9) |
$-\,2^{12}\cdot 3^{27}\cdot 5^{8}\cdot 137^{12}$ |
$4$ |
$448.231381711$ |
$818.1932124074161$ |
|
|
|
$C_3^2:S_3$ (as 18T24) |
$[3]$ |
$[3]$ |
$6$ |
$8$ |
$159078827156081020$ |
$0$ |
$137$ |
| 18.0.677...203.1 |
$x^{18} + 22 x^{16} + 5123 x^{14} + 190443 x^{12} - 4281567 x^{10} - 64033813 x^{8} + 2317543201 x^{6} - 1388120511 x^{4} - 79706858886 x^{2} + 1344164574483$ |
$18$ |
(0, 9) |
$-\,3^{9}\cdot 31^{12}\cdot 137^{12}$ |
$3$ |
$454.236610694$ |
$454.23661069402925$ |
|
✓ |
|
$S_3 \times C_3$ (as 18T3) |
trivial |
trivial |
$6$ |
$8$ |
$1159831218324532000$ |
$0$ |
$137$ |
| 18.6.478...000.1 |
$x^{18} - 9 x^{17} + 199 x^{16} + 1672 x^{15} - 24014 x^{14} + 538474 x^{13} - 4979400 x^{12} + 9746476 x^{11} + 34307513 x^{10} - 6363703971 x^{9} + 48715534889 x^{8} - 270954554452 x^{7} + 2637123450936 x^{6} + 7805929179998 x^{5} - 67547824100486 x^{4} - 479892245158660 x^{3} + 655326484877599 x^{2} + 4276397853662235 x + 13853779377943345$ |
$18$ |
(6, 6) |
$2^{12}\cdot 3^{22}\cdot 5^{13}\cdot 17^{8}\cdot 137^{12}$ |
$5$ |
$1819.6719442538188$ |
$7992.150007686705$ |
|
|
|
$C_3^2:D_6$ (as 18T52) |
not computed |
not computed |
$2$ |
$11$ |
|
|
$137$ |
| 18.6.182...000.1 |
$x^{18} - 7 x^{17} - 1270 x^{16} + 27327 x^{15} - 539570 x^{14} - 1765817 x^{13} + 715346969 x^{12} - 15180490590 x^{11} + 191387571696 x^{10} - 3079633873400 x^{9} + 30603146449760 x^{8} - 326837621309920 x^{7} + 3748742204203840 x^{6} - 22178257412455680 x^{5} + 378890995640378880 x^{4} - 1569194505105809408 x^{3} + 27431207687819165696 x^{2} - 103785057364796293120 x + 757197959525749080064$ |
$18$ |
(6, 6) |
$2^{8}\cdot 3^{22}\cdot 5^{15}\cdot 19^{8}\cdot 137^{12}$ |
$5$ |
$1959.8806488804462$ |
$8607.296160260985$ |
|
|
|
$C_3^2:D_6$ (as 18T52) |
$[3, 3, 3]$ |
$[3, 3, 3]$ |
$2$ |
$11$ |
$20973385237684180000000$ |
$6$ |
$137$ |
| 18.6.100...000.1 |
$x^{18} + 82 x^{16} - 3060 x^{15} - 11564 x^{14} - 346800 x^{13} - 2316260 x^{12} + 16062960 x^{11} - 353061200 x^{10} + 6558783600 x^{9} + 25784011600 x^{8} + 73174392000 x^{7} + 2103214780000 x^{6} - 38601776784000 x^{5} - 48981151520000 x^{4} + 903902649360000 x^{3} + 2312103293560000 x^{2} + 5766379647600000 x + 12095850965400000$ |
$18$ |
(6, 6) |
$2^{33}\cdot 3^{22}\cdot 5^{13}\cdot 17^{8}\cdot 137^{12}$ |
$5$ |
$4085.0253955964813$ |
$17941.77013604758$ |
|
|
|
$C_3^2:D_6$ (as 18T52) |
not computed |
not computed |
$2$ |
$11$ |
|
|
$137$ |
