| 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 |
| 16.8.115...000.1 |
$x^{16} - 60 x^{14} + 732 x^{12} + 11080 x^{10} - 175849 x^{8} - 547080 x^{6} + 5112410 x^{4} + 15657100 x^{2} + 21025$ |
$16$ |
[8,4] |
$2^{48}\cdot 5^{10}\cdot 29^{10}$ |
$3$ |
$179.45063042673704$ |
$364.5396213410056$ |
|
|
? |
$C_2^5.D_4$ (as 16T544) |
$[2, 4]$ |
$[2, 2, 2, 4]$ |
$2$ |
$11$ |
$349002249514.1265$ |
| 27.3.117...000.1 |
$x^{27} + 180 x^{21} + 1800 x^{19} - 9240 x^{18} - 184950 x^{15} + 86400 x^{13} + 138600 x^{12} - 486000 x^{11} - 1821600 x^{10} - 4056900 x^{9} + 5751000 x^{7} + 7425000 x^{6} + 1944000 x^{5} + 1188000 x^{4} - 17778375 x^{3} + 2178000 x - 1331000$ |
$27$ |
[3,12] |
$2^{70}\cdot 3^{34}\cdot 5^{24}$ |
$3$ |
$100.5944835280339$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$14$ |
$632270644265912400$ |
| 27.3.122...000.1 |
$x^{27} - 6444 x^{21} - 27000 x^{19} - 81144 x^{18} + 3805002 x^{15} + 46396800 x^{13} - 43574328 x^{12} - 109350000 x^{11} + 239954400 x^{10} - 5747266980 x^{9} + 19495258200 x^{7} - 9492909048 x^{6} - 15658920000 x^{5} + 5602413600 x^{4} - 26876890503 x^{3} - 2519521200 x - 901428696$ |
$27$ |
[3,12] |
$2^{70}\cdot 3^{60}\cdot 5^{12}$ |
$3$ |
$141.70035135179322$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
$[3]$ |
$[3]$ |
$2$ |
$14$ |
$15946304910053687000$ |
| 27.3.193...536.2 |
$x^{27} - 972 x^{21} + 3240 x^{19} - 38808 x^{18} - 610902 x^{15} - 839808 x^{13} - 3143448 x^{12} - 1574640 x^{11} - 13771296 x^{10} - 7956036 x^{9} - 8030664 x^{7} + 25596648 x^{6} - 34012224 x^{5} - 48498912 x^{4} - 63171495 x^{3} + 69155856 x - 98611128$ |
$27$ |
[3,12] |
$2^{70}\cdot 3^{78}$ |
$2$ |
$144.14461692570003$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
$[9]$ |
$[9]$ |
$2$ |
$14$ |
$7522732427772556000$ |
| 27.7.383...000.1 |
$x^{27} - 9 x^{26} + 15 x^{25} + 225 x^{24} - 3510 x^{23} + 20886 x^{22} + 5866 x^{21} - 646770 x^{20} + 2284215 x^{19} + 3031145 x^{18} - 34366251 x^{17} + 41634939 x^{16} + 247814950 x^{15} - 639481710 x^{14} - 2339954610 x^{13} + 9860011346 x^{12} + 7028475801 x^{11} - 72605056905 x^{10} + 26719309095 x^{9} + 241150841745 x^{8} + 32140560108 x^{7} - 2099267919292 x^{6} + 3829503194340 x^{5} - 1141671272100 x^{4} + 2099477945700 x^{3} - 6265891409220 x^{2} - 5810885182020 x + 1175091492900$ |
$27$ |
[7,10] |
$2^{70}\cdot 3^{28}\cdot 5^{46}$ |
$3$ |
$292.46798593417213$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
trivial |
trivial |
$2$ |
$16$ |
$6617062149659400000000000$ |
| 27.3.473...000.4 |
$x^{27} - 9 x^{26} - 9 x^{25} - 207 x^{24} + 4554 x^{23} + 4446 x^{22} - 123174 x^{21} - 595602 x^{20} + 2699262 x^{19} + 21217794 x^{18} - 19512450 x^{17} - 450045054 x^{16} - 552214170 x^{15} + 5895691074 x^{14} + 18042425262 x^{13} - 40829771574 x^{12} - 248579493606 x^{11} + 15216716934 x^{10} + 1908486447330 x^{9} + 2021878167462 x^{8} - 7847788083120 x^{7} - 14219552157912 x^{6} + 14813366626764 x^{5} + 33214672439820 x^{4} - 22530031062165 x^{3} - 25209629231211 x^{2} + 104301124750305 x + 123824201260823$ |
$27$ |
[3,12] |
$2^{70}\cdot 3^{78}\cdot 5^{12}$ |
$3$ |
$294.74860856763604$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
not computed |
not computed |
$2$ |
$14$ |
|
| 27.3.739...000.1 |
$x^{27} + 108 x^{25} + 4860 x^{23} - 1296 x^{22} + 121176 x^{21} - 81648 x^{20} + 1874502 x^{19} - 43455888 x^{18} + 19105632 x^{17} - 1570875120 x^{16} + 929695644 x^{15} - 22245055920 x^{14} + 21515637960 x^{13} - 151991465040 x^{12} + 177042520449 x^{11} - 478870127280 x^{10} - 38023433783532 x^{9} - 144362680272 x^{8} - 327249555954912 x^{7} - 153687847931136 x^{6} - 942744685555584 x^{5} - 1485071721856512 x^{4} - 1603805039407872 x^{3} - 2820400383909888 x^{2} - 2053295366446080 x - 120547873823772672$ |
$27$ |
[3,12] |
$2^{70}\cdot 3^{78}\cdot 5^{18}$ |
$3$ |
$421.4814167586454$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
not computed |
not computed |
$2$ |
$14$ |
|
| 27.3.739...000.11 |
$x^{27} - 36 x^{25} - 1305 x^{24} + 1809 x^{23} + 35262 x^{22} + 100764 x^{21} + 377595 x^{20} + 836190 x^{19} + 2728926 x^{18} - 49628754 x^{17} - 331644402 x^{16} - 1336717215 x^{15} - 9562333986 x^{14} - 2069838360 x^{13} + 71698250907 x^{12} + 442661193270 x^{11} + 1630540589778 x^{10} + 3915767379546 x^{9} + 7686816794910 x^{8} - 32861139532407 x^{7} + 45782288303202 x^{6} + 71054877286008 x^{5} + 151356435918219 x^{4} + 5068200693137301 x^{3} + 5939904479004018 x^{2} + 1167638461385346 x + 13348289204316327$ |
$27$ |
[3,12] |
$2^{70}\cdot 3^{78}\cdot 5^{18}$ |
$3$ |
$421.4814167586454$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
not computed |
not computed |
$2$ |
$14$ |
|
| 27.3.176...000.2 |
$x^{27} + 432000 x^{19} - 8601600 x^{18} - 103680000 x^{15} - 27993600000 x^{11} - 406978560000 x^{10} - 1667235840000 x^{9} + 895795200000 x^{7} + 4246732800000 x^{6} - 15925248000000 x^{3} + 452984832000000 x - 1073741824000000$ |
$27$ |
[3,12] |
$2^{70}\cdot 3^{70}\cdot 5^{24}$ |
$3$ |
$435.2470519308077$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
not computed |
not computed |
$2$ |
$14$ |
|
| 27.3.115...000.9 |
$x^{27} - 12960 x^{21} - 32400 x^{19} - 947520 x^{18} - 37810800 x^{15} + 111974400 x^{13} - 1023321600 x^{12} - 157464000 x^{11} + 3362342400 x^{10} - 19811001600 x^{9} + 60151248000 x^{7} - 208319040000 x^{6} - 45349632000 x^{5} + 157883904000 x^{4} - 1192241592000 x^{3} - 412252416000 x - 1435249152000$ |
$27$ |
[3,12] |
$2^{70}\cdot 3^{78}\cdot 5^{24}$ |
$3$ |
$602.7054225503165$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
not computed |
not computed |
$2$ |
$14$ |
|
| 27.3.182...000.3 |
$x^{27} - 360 x^{25} + 54000 x^{23} - 9216 x^{22} - 4446720 x^{21} + 1935360 x^{20} + 223891200 x^{19} - 147947520 x^{18} - 7264733184 x^{17} + 5311365120 x^{16} + 157728187440 x^{15} - 91269365760 x^{14} - 2382138768960 x^{13} + 545341833216 x^{12} + 25423077070080 x^{11} + 6622870118400 x^{10} - 187627429063680 x^{9} - 158032055255040 x^{8} + 882954372464640 x^{7} + 1363404496896000 x^{6} - 2063174288670720 x^{5} - 5766067448709120 x^{4} - 622299537144000 x^{3} + 9381033208184832 x^{2} + 10296524915343360 x + 3522514072371200$ |
$27$ |
[3,12] |
$2^{70}\cdot 3^{60}\cdot 5^{38}$ |
$3$ |
$667.5028948393045$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
not computed |
not computed |
$2$ |
$14$ |
|
| 27.3.177...000.6 |
$x^{27} - 360 x^{25} + 54000 x^{23} - 9216 x^{22} - 4446720 x^{21} + 1935360 x^{20} + 223891200 x^{19} - 147947520 x^{18} - 7264733184 x^{17} + 5311365120 x^{16} + 159244507440 x^{15} - 91269365760 x^{14} - 2497379088960 x^{13} + 545341833216 x^{12} + 28698328270080 x^{11} + 7709768294400 x^{10} - 231976756423680 x^{9} - 199955270615040 x^{8} + 1167100609904640 x^{7} + 1878904774656000 x^{6} - 2532714300702720 x^{5} - 7902598720389120 x^{4} - 3861559365624000 x^{3} + 9381033208184832 x^{2} + 13476478435983360 x + 6066476888883200$ |
$27$ |
[3,12] |
$2^{70}\cdot 3^{60}\cdot 5^{48}$ |
$3$ |
$1211.5209622360157$ |
|
|
|
|
$\SO(5,3)$ (as 27T1161) |
not computed |
not computed |
$2$ |
$14$ |
|
