| 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 |
| 21.3.717...392.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 42 x^{18} + 77 x^{17} - 126 x^{16} + 168 x^{15} - 213 x^{14} + 266 x^{13} - 280 x^{12} + 259 x^{11} - 217 x^{10} + 133 x^{9} - 42 x^{8} - 53 x^{7} + 126 x^{6} - 112 x^{5} + 7 x^{4} + 63 x^{3} - 49 x^{2} + 14 x - 1$ |
$21$ |
[3,9] |
$-\,2^{18}\cdot 7^{23}$ |
$2$ |
$15.2619237063$ |
$15.985663254345006$ |
|
|
? |
$F_7$ (as 21T4) |
trivial |
$2$ |
$11$ |
$10096.3106241$ |
| 21.3.928...699.1 |
$x^{21} - 5 x^{20} + 11 x^{19} - 13 x^{18} + 7 x^{17} + 4 x^{16} - 16 x^{15} + 13 x^{14} - 6 x^{13} + 2 x^{12} - 2 x^{11} + 21 x^{10} - 28 x^{8} + 29 x^{7} - 16 x^{6} - 18 x^{5} + 17 x^{4} - 9 x^{3} - 2 x^{2} + 3 x - 1$ |
$21$ |
[3,9] |
$-\,11^{9}\cdot 13^{14}$ |
$2$ |
$15.4504115563$ |
$18.336871607339905$ |
|
|
? |
$F_7$ (as 21T4) |
trivial |
$2$ |
$11$ |
$10371.33223$ |
| 21.3.378...943.3 |
$x^{21} - x^{14} - 9 x^{7} + 1$ |
$21$ |
[3,9] |
$-\,7^{35}$ |
$1$ |
$25.6151399702$ |
$26.829842007890413$ |
|
|
|
$F_7$ (as 21T4) |
trivial |
$2$ |
$11$ |
$5449377.399032222$ |
| 21.21.941...237.1 |
$x^{21} - 31 x^{19} - 3 x^{18} + 362 x^{17} + 32 x^{16} - 2119 x^{15} - 158 x^{14} + 6826 x^{13} + 676 x^{12} - 12509 x^{11} - 2021 x^{10} + 12809 x^{9} + 3175 x^{8} - 6710 x^{7} - 2254 x^{6} + 1442 x^{5} + 571 x^{4} - 97 x^{3} - 44 x^{2} + 2 x + 1$ |
$21$ |
[21,0] |
$7^{14}\cdot 173^{9}$ |
$2$ |
$33.3089780981$ |
$48.13065200407494$ |
|
|
? |
$F_7$ (as 21T4) |
trivial |
$2$ |
$20$ |
$1629233271.9$ |
| 21.21.240...312.1 |
$x^{21} - 2 x^{20} - 32 x^{19} + 51 x^{18} + 432 x^{17} - 473 x^{16} - 3214 x^{15} + 1767 x^{14} + 14108 x^{13} - 305 x^{12} - 35382 x^{11} - 15763 x^{10} + 43350 x^{9} + 37753 x^{8} - 14531 x^{7} - 24311 x^{6} - 2788 x^{5} + 4492 x^{4} + 1052 x^{3} - 276 x^{2} - 56 x + 8$ |
$21$ |
[21,0] |
$2^{27}\cdot 7^{14}\cdot 31^{9}$ |
$3$ |
$38.8680068717$ |
$57.626803948274514$ |
|
|
|
$F_7$ (as 21T4) |
trivial |
$2$ |
$20$ |
$12111978767.0$ |
| 21.3.261...703.2 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 32 x^{16} + 201 x^{15} - 718 x^{14} + 847 x^{13} + 988 x^{12} - 1701 x^{11} - 6256 x^{10} + 18797 x^{9} - 15492 x^{8} - 45562 x^{7} + 150977 x^{6} - 83972 x^{5} - 198072 x^{4} + 215936 x^{3} + 70896 x^{2} - 133952 x + 19264$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 13^{18}$ |
$2$ |
$43.5447270343$ |
$45.60967255955715$ |
|
|
|
$F_7$ (as 21T4) |
trivial |
$2$ |
$11$ |
$3123124290.4668374$ |
| 21.3.489...327.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 52 x^{16} - 373 x^{15} + 248 x^{14} - 3577 x^{13} + 42918 x^{12} - 170639 x^{11} + 225822 x^{10} + 43556 x^{9} + 105237 x^{8} - 3230632 x^{7} + 5161290 x^{6} + 1287601 x^{5} + 2320101 x^{4} - 14827477 x^{3} + 4310250 x^{2} + 1634962 x + 569387$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 29^{18}$ |
$2$ |
$86.6185646862$ |
$90.72612557219027$ |
|
|
|
$F_7$ (as 21T4) |
$[7, 7]$ |
$2$ |
$11$ |
$72630093220.36017$ |
| 21.3.249...847.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 316 x^{17} - 375 x^{16} + 5 x^{15} - 1019 x^{14} + 20797 x^{13} + 115088 x^{12} + 265993 x^{11} + 283880 x^{10} + 220817 x^{9} + 3096540 x^{8} + 13444516 x^{7} + 20299735 x^{6} - 8975624 x^{5} - 86530794 x^{4} - 152378359 x^{3} - 129757565 x^{2} - 50056244 x - 6446237$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 41^{18}$ |
$2$ |
$116.55023343$ |
$122.077191557991$ |
|
|
|
$F_7$ (as 21T4) |
trivial |
$2$ |
$11$ |
$115224462320340.23$ |
| 21.3.587...343.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 330 x^{17} - 2629 x^{16} + 9945 x^{15} - 21697 x^{14} + 61691 x^{13} - 337490 x^{12} + 1367751 x^{11} - 3880770 x^{10} + 9707987 x^{9} - 27063618 x^{8} + 86251922 x^{7} - 249930855 x^{6} + 516125610 x^{5} - 636836088 x^{4} + 367896053 x^{3} + 18430223 x^{2} - 116740442 x + 35136283$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 43^{18}$ |
$2$ |
$121.406741733$ |
$127.16400157052036$ |
|
|
|
$F_7$ (as 21T4) |
$[7]$ |
$2$ |
$11$ |
$17109387227548.602$ |
| 21.3.426...647.1 |
$x^{21} - 2067 x^{14} - 766753 x^{7} - 62748517$ |
$21$ |
[3,9] |
$-\,7^{35}\cdot 13^{18}$ |
$2$ |
$230.836946607$ |
$241.78352389457936$ |
|
|
|
$F_7$ (as 21T4) |
trivial |
$2$ |
$11$ |
$53577224642982600$ |
| 21.3.426...647.2 |
$x^{21} - 2171 x^{14} + 1219335 x^{7} - 62748517$ |
$21$ |
[3,9] |
$-\,7^{35}\cdot 13^{18}$ |
$2$ |
$230.836946607$ |
$241.78352389457936$ |
|
|
|
$F_7$ (as 21T4) |
trivial |
$2$ |
$11$ |
$50323767605348790$ |
| 21.3.426...647.3 |
$x^{21} - 325 x^{14} - 221897 x^{7} + 62748517$ |
$21$ |
[3,9] |
$-\,7^{35}\cdot 13^{18}$ |
$2$ |
$230.836946607$ |
$241.78352389457936$ |
|
|
|
$F_7$ (as 21T4) |
$[2, 2, 2]$ |
$2$ |
$11$ |
$8501667076335819.0$ |
| 21.3.209...503.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 3384 x^{16} - 22661 x^{15} + 85004 x^{14} - 409885 x^{13} - 188418 x^{12} - 2604021 x^{11} - 6603224 x^{10} + 30173537 x^{9} - 75034142 x^{8} - 399264676 x^{7} + 477537789 x^{6} + 583377214 x^{5} - 1978514300 x^{4} + 979864648 x^{3} + 3374286608 x^{2} - 5170051936 x + 2015987008$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 113^{18}$ |
$2$ |
$277.912773094$ |
$291.09174506868976$ |
|
|
|
$F_7$ (as 21T4) |
$[7]$ |
$2$ |
$11$ |
$104794948949991400$ |
| 21.3.171...183.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} - 860 x^{17} + 4693 x^{16} - 12259 x^{15} - 14067 x^{14} + 37681 x^{13} - 3477004 x^{12} + 389109 x^{11} + 23235788 x^{10} - 37794391 x^{9} + 399984668 x^{8} + 105257496 x^{7} - 6303984553 x^{6} + 3887930312 x^{5} + 9054965094 x^{4} - 95684899751 x^{3} + 28879725259 x^{2} + 175769815592 x - 322494640909$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 127^{18}$ |
$2$ |
$307.176025635$ |
$321.7426977172085$ |
|
|
|
$F_7$ (as 21T4) |
$[7]$ |
$2$ |
$11$ |
$268187551415576800$ |
| 21.3.237...663.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 9692 x^{16} + 37133 x^{15} - 111234 x^{14} - 306733 x^{13} + 2227436 x^{12} - 15927429 x^{11} + 48390428 x^{10} - 36546963 x^{9} - 117158588 x^{8} - 3295825882 x^{7} + 13211621361 x^{6} - 7267451240 x^{5} - 59139292492 x^{4} + 73337235104 x^{3} + 87604194608 x^{2} - 76802664064 x - 146043093056$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 167^{18}$ |
$2$ |
$388.429791287$ |
$406.84961876184303$ |
|
|
|
$F_7$ (as 21T4) |
trivial |
$2$ |
$11$ |
$20918292900910133000$ |
| 21.3.797...823.1 |
$x^{21} - 25955 x^{14} - 81483649 x^{7} - 17249876309$ |
$21$ |
[3,9] |
$-\,7^{35}\cdot 29^{18}$ |
$2$ |
$459.177639944$ |
$480.95241906203387$ |
|
|
|
$F_7$ (as 21T4) |
$[7]$ |
$2$ |
$11$ |
$11544690986222539000$ |
| 21.3.159...767.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 12232 x^{16} - 36395 x^{15} + 28738 x^{14} - 856933 x^{13} - 5891416 x^{12} + 24379691 x^{11} - 5679616 x^{10} + 73289981 x^{9} - 350840512 x^{8} - 12471465894 x^{7} + 5267658761 x^{6} + 83230171864 x^{5} + 22083422172 x^{4} - 43889160896 x^{3} - 214646951456 x^{2} - 858652083072 x + 745947224512$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 211^{18}$ |
$2$ |
$474.645195654$ |
$497.153465646847$ |
|
|
|
$F_7$ (as 21T4) |
$[7, 7, 7]$ |
$2$ |
$11$ |
$482362851034983940$ |
| 21.3.432...983.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 6696 x^{16} + 39331 x^{15} - 66028 x^{14} - 967981 x^{13} + 7864242 x^{12} - 17206833 x^{11} - 16939676 x^{10} + 352134365 x^{9} - 809756522 x^{8} - 16153054924 x^{7} + 78811833717 x^{6} + 8157442846 x^{5} - 535267042184 x^{4} + 402334865128 x^{3} + 1100178148832 x^{2} - 2037042160384 x - 256310093312$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 223^{18}$ |
$2$ |
$497.690928127$ |
$521.2920556345857$ |
|
|
|
$F_7$ (as 21T4) |
trivial |
$2$ |
$11$ |
$327908201372163200000$ |
| 21.3.363...207.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} - 1728 x^{17} + 8277 x^{16} - 26511 x^{15} + 228637 x^{14} - 2299843 x^{13} + 43252952 x^{12} - 358447607 x^{11} + 1629066924 x^{10} - 4992150763 x^{9} + 6624085180 x^{8} - 9335801376 x^{7} + 136223349179 x^{6} - 270952034724 x^{5} + 131571710354 x^{4} - 3552798817939 x^{3} + 6818751637159 x^{2} + 1165451658700 x + 7016855542291$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 251^{18}$ |
$2$ |
$550.795235191$ |
$576.914635488129$ |
|
|
|
$F_7$ (as 21T4) |
$[2, 2, 2]$ |
$2$ |
$11$ |
$136686531300247910000$ |
| 21.3.589...343.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 17000 x^{16} + 76879 x^{15} + 25168 x^{14} - 3960985 x^{13} - 7944166 x^{12} + 40651037 x^{11} + 81259502 x^{10} - 506512648 x^{9} - 2778664855 x^{8} - 84195862092 x^{7} + 400745644202 x^{6} + 655623718393 x^{5} - 7349761489371 x^{4} + 529795117999 x^{3} + 53152215171034 x^{2} - 12893672008602 x - 131863811763929$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 293^{18}$ |
$2$ |
$628.904851644$ |
$658.7283078391205$ |
|
|
|
$F_7$ (as 21T4) |
$[7, 7]$ |
$2$ |
$11$ |
$77992079112627040000$ |
| 21.3.956...007.1 |
$x^{21} - 89483 x^{14} + 103438607 x^{7} + 271818611107$ |
$21$ |
[3,9] |
$-\,7^{35}\cdot 43^{18}$ |
$2$ |
$643.594838408$ |
$674.1149121846441$ |
|
|
|
$F_7$ (as 21T4) |
$[7]$ |
$2$ |
$11$ |
$515055862320607300000$ |
| 21.3.605...727.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 5335 x^{17} - 68226 x^{16} + 370487 x^{15} - 976238 x^{14} + 9992983 x^{13} - 167147434 x^{12} + 1254295567 x^{11} - 5119024426 x^{10} + 16406088995 x^{9} - 124794841242 x^{8} + 1235524808890 x^{7} - 7765632654107 x^{6} + 28259771083054 x^{5} - 48713312542188 x^{4} - 12644866212328 x^{3} + 193183257790096 x^{2} - 276127419440096 x + 122588921675584$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 379^{18}$ |
$2$ |
$784.131976902$ |
$821.3164979036171$ |
|
|
|
$F_7$ (as 21T4) |
$[7, 7, 7, 7, 7]$ |
$2$ |
$11$ |
$2564983668887593500$ |
| 21.3.639...047.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 42220 x^{16} - 238849 x^{15} + 507776 x^{14} - 9461221 x^{13} + 30949290 x^{12} - 118397153 x^{11} + 849128436 x^{10} + 3360629141 x^{9} - 17534305986 x^{8} - 639064890904 x^{7} - 84897810087 x^{6} + 8605795373014 x^{5} + 21486789774576 x^{4} - 76273061749528 x^{3} - 178475909056704 x^{2} + 177581658222976 x + 629094860366336$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 491^{18}$ |
$2$ |
$978.967476347$ |
$1025.391340896248$ |
|
|
|
$F_7$ (as 21T4) |
not computed |
$2$ |
$11$ |
|
| 21.3.280...263.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 30908 x^{16} - 227299 x^{15} + 331614 x^{14} - 11256581 x^{13} - 100744748 x^{12} + 250586819 x^{11} - 1122038804 x^{10} + 8083253229 x^{9} - 7342418196 x^{8} - 1136578032834 x^{7} + 596900678473 x^{6} + 16991239839504 x^{5} - 27591656221276 x^{4} - 9543020500384 x^{3} + 370902066797360 x^{2} - 862711219086080 x - 660800505832768$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 13^{18}\cdot 41^{18}$ |
$3$ |
$1050.32024196$ |
$1100.1277440755782$ |
|
|
|
$F_7$ (as 21T4) |
$[14, 14, 14]$ |
$2$ |
$11$ |
$445424233834374960000$ |
| 21.3.132...687.1 |
$x^{21} - 57851 x^{14} + 30304711 x^{7} + 27136050989627$ |
$21$ |
[3,9] |
$-\,7^{35}\cdot 83^{18}$ |
$2$ |
$1130.89116851$ |
$1184.519444924393$ |
|
|
|
$F_7$ (as 21T4) |
not computed |
$2$ |
$11$ |
|
| 21.3.243...807.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 34864 x^{16} - 57451 x^{15} + 891810 x^{14} - 16216613 x^{13} + 111348616 x^{12} - 1168389061 x^{11} + 7475278120 x^{10} - 2937883611 x^{9} - 48950465208 x^{8} - 1285942710366 x^{7} + 6911521990801 x^{6} - 1175582555784 x^{5} - 35643871943428 x^{4} - 19183703267008 x^{3} - 9595175213344 x^{2} - 215779097824640 x - 84881631489088$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 601^{18}$ |
$2$ |
$1164.1780037$ |
$1219.3847835596653$ |
|
|
|
$F_7$ (as 21T4) |
not computed |
$2$ |
$11$ |
|
| 21.3.390...063.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 53056 x^{16} - 387753 x^{15} + 1715360 x^{14} - 20945477 x^{13} - 106972970 x^{12} - 574361697 x^{11} - 3215159260 x^{10} + 18233746589 x^{9} - 44219027574 x^{8} - 1367906097928 x^{7} - 307347689851 x^{6} + 10689969421306 x^{5} - 11010699170856 x^{4} - 17385833768968 x^{3} + 157072709296608 x^{2} - 428208522920192 x + 293698546559488$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 617^{18}$ |
$2$ |
$1190.69349917$ |
$1247.1576770544202$ |
|
|
|
$F_7$ (as 21T4) |
not computed |
$2$ |
$11$ |
|
| 21.3.205...087.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 66140 x^{16} + 197181 x^{15} + 519746 x^{14} - 38250611 x^{13} + 728885410 x^{12} - 5190026793 x^{11} + 29182239326 x^{10} - 68615906434 x^{9} + 126081016179 x^{8} - 6211611183364 x^{7} + 23232341812268 x^{6} + 42625508333359 x^{5} + 56176184509851 x^{4} - 383224347675253 x^{3} - 3177193347653934 x^{2} - 1410935175422024 x + 6295917838747831$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 769^{18}$ |
$2$ |
$1438.06373091$ |
$1506.258515181767$ |
|
|
|
$F_7$ (as 21T4) |
not computed |
$2$ |
$11$ |
|
| 21.3.279...167.1 |
$x^{21} - 1173353 x^{14} - 82982043713 x^{7} - 532875860165503$ |
$21$ |
[3,9] |
$-\,7^{35}\cdot 127^{18}$ |
$2$ |
$1628.38489658$ |
$1705.6049490343119$ |
|
|
|
$F_7$ (as 21T4) |
not computed |
$2$ |
$11$ |
|
| 21.3.698...103.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 150952 x^{16} + 1091081 x^{15} - 1885902 x^{14} - 82474679 x^{13} + 1782947986 x^{12} - 8271270553 x^{11} + 14713958422 x^{10} + 36609721794 x^{9} - 157374528081 x^{8} - 29032774648716 x^{7} + 146378771353312 x^{6} + 403041892117935 x^{5} - 1511275454987137 x^{4} + 2913800887665479 x^{3} - 25119733760908906 x^{2} - 90692100544127768 x + 135973981979723027$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 1063^{18}$ |
$2$ |
$1898.0093799$ |
$1988.0153632359043$ |
|
|
|
$F_7$ (as 21T4) |
not computed |
$2$ |
$11$ |
|
| 21.3.146...047.1 |
$x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 214036 x^{16} + 1591903 x^{15} - 8455612 x^{14} - 78804229 x^{13} + 2442806710 x^{12} - 18581839787 x^{11} + 108000335718 x^{10} - 222335916920 x^{9} + 284300561201 x^{8} - 44833968691560 x^{7} + 248434219855942 x^{6} + 39614286391869 x^{5} - 1260636204184623 x^{4} + 2502989432053803 x^{3} - 12369631255580718 x^{2} - 62218052737609482 x - 26412654578702301$ |
$21$ |
[3,9] |
$-\,7^{17}\cdot 1259^{18}$ |
$2$ |
$2194.27925016$ |
$2298.3347220218943$ |
|
|
|
$F_7$ (as 21T4) |
not computed |
$2$ |
$11$ |
|
