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 |
8.8.282300416.1 |
$x^{8} - 4 x^{7} + 14 x^{5} - 8 x^{4} - 12 x^{3} + 7 x^{2} + 2 x - 1$ |
$8$ |
[8,0] |
$2^{12}\cdot 41^{3}$ |
$2$ |
$11.3851547954$ |
$45.82820910616052$ |
|
|
✓ |
$C_4\wr C_2$ (as 8T17) |
trivial |
$2$ |
$7$ |
$22.4468700736$ |
8.8.309593125.1 |
$x^{8} - 4 x^{7} - x^{6} + 17 x^{5} - 5 x^{4} - 23 x^{3} + 6 x^{2} + 9 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 19\cdot 29^{2}\cdot 31$ |
$4$ |
$11.5172531693$ |
$292.24133862272123$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$23.6967890277$ |
8.8.324000000.1 |
$x^{8} - 7 x^{6} + 14 x^{4} - 8 x^{2} + 1$ |
$8$ |
[8,0] |
$2^{8}\cdot 3^{4}\cdot 5^{6}$ |
$3$ |
$11.5829218529$ |
$11.582921852882691$ |
|
✓ |
✓ |
$C_4\times C_2$ (as 8T2) |
trivial |
$2$ |
$7$ |
$24.3884064258$ |
8.8.410338673.1 |
$x^{8} - x^{7} - 7 x^{6} + 6 x^{5} + 15 x^{4} - 10 x^{3} - 10 x^{2} + 4 x + 1$ |
$8$ |
[8,0] |
$17^{7}$ |
$1$ |
$11.9300648987$ |
$11.93006489865314$ |
|
✓ |
✓ |
$C_8$ (as 8T1) |
trivial |
$2$ |
$7$ |
$28.437595417$ |
8.8.432640000.1 |
$x^{8} - 2 x^{7} - 7 x^{6} + 16 x^{5} + 4 x^{4} - 18 x^{3} + 2 x^{2} + 4 x - 1$ |
$8$ |
[8,0] |
$2^{12}\cdot 5^{4}\cdot 13^{2}$ |
$3$ |
$12.0092485618$ |
$22.80350850198276$ |
|
|
✓ |
$Q_8:C_2$ (as 8T11) |
trivial |
$2$ |
$7$ |
$28.9890220694$ |
8.8.442050625.1 |
$x^{8} - 2 x^{7} - 12 x^{6} + 26 x^{5} + 17 x^{4} - 36 x^{3} - 5 x^{2} + 11 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 29^{4}$ |
$2$ |
$12.0415945788$ |
$12.041594578792296$ |
|
✓ |
? |
$D_4$ (as 8T4) |
trivial |
$2$ |
$7$ |
$29.6385151004$ |
8.8.456768125.1 |
$x^{8} - 2 x^{7} - 7 x^{6} + 11 x^{5} + 14 x^{4} - 18 x^{3} - 8 x^{2} + 9 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 11\cdot 29^{2}\cdot 79$ |
$4$ |
$12.0909931103$ |
$354.97182986823054$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$30.3398226576$ |
8.8.483345053.1 |
$x^{8} - x^{7} - 7 x^{6} + 4 x^{5} + 15 x^{4} - 3 x^{3} - 9 x^{2} + 1$ |
$8$ |
[8,0] |
$483345053$ |
$1$ |
$12.1767715918$ |
$21985.109801863626$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$31.4056495019$ |
8.8.494613125.1 |
$x^{8} - x^{7} - 7 x^{6} + 4 x^{5} + 13 x^{4} - 4 x^{3} - 7 x^{2} + x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 29^{2}\cdot 941$ |
$3$ |
$12.2118990229$ |
$369.3846233940985$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$31.4375526025$ |
8.8.582918125.1 |
$x^{8} - 2 x^{7} - 6 x^{6} + 9 x^{5} + 11 x^{4} - 9 x^{3} - 6 x^{2} + 2 x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 29^{2}\cdot 1109$ |
$3$ |
$12.4652486189$ |
$401.00498750015566$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$35.7205035025$ |
8.8.656505625.1 |
$x^{8} - 3 x^{7} - 4 x^{6} + 13 x^{5} + 5 x^{4} - 13 x^{3} - 4 x^{2} + 3 x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 29^{2}\cdot 1249$ |
$3$ |
$12.6518724942$ |
$425.5643312120977$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$37.2825779024$ |
8.8.661518125.1 |
$x^{8} - x^{7} - 7 x^{6} + 5 x^{5} + 15 x^{4} - 7 x^{3} - 10 x^{2} + 2 x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 439\cdot 2411$ |
$3$ |
$12.6639071738$ |
$2300.4662570878972$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$7$ |
$37.8654262114$ |
8.8.707295133.1 |
$x^{8} - 8 x^{6} - 2 x^{5} + 19 x^{4} + 7 x^{3} - 13 x^{2} - 4 x + 1$ |
$8$ |
[8,0] |
$9973\cdot 70921$ |
$2$ |
$12.7702701211$ |
$26595.02083097511$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$39.2596652535$ |
8.8.733968125.1 |
$x^{8} - 2 x^{7} - 6 x^{6} + 10 x^{5} + 11 x^{4} - 11 x^{3} - 7 x^{2} + 2 x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 11\cdot 106759$ |
$3$ |
$12.8294976034$ |
$2423.1683804473846$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$7$ |
$40.3475819574$ |
8.8.740605625.1 |
$x^{8} - x^{7} - 9 x^{6} + 8 x^{5} + 21 x^{4} - 12 x^{3} - 14 x^{2} + 4 x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 29^{2}\cdot 1409$ |
$3$ |
$12.8439431844$ |
$452.0011061933367$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$40.3991503507$ |
8.8.803680625.1 |
$x^{8} - 2 x^{7} - 9 x^{6} + 12 x^{5} + 22 x^{4} - 24 x^{3} - 14 x^{2} + 14 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 11\cdot 29^{2}\cdot 139$ |
$4$ |
$12.9758386715$ |
$470.8556041930477$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$42.7587953365$ |
8.8.852038125.1 |
$x^{8} - 10 x^{6} - 5 x^{5} + 17 x^{4} + 5 x^{3} - 10 x^{2} + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 29^{2}\cdot 1621$ |
$3$ |
$13.0709567759$ |
$484.81439747598256$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$44.7501704817$ |
8.8.877268125.1 |
$x^{8} - 3 x^{7} - 6 x^{6} + 20 x^{5} + 5 x^{4} - 25 x^{3} - x^{2} + 7 x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 29^{2}\cdot 1669$ |
$3$ |
$13.1187224452$ |
$491.9400369963803$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$45.8191285307$ |
8.8.898293125.1 |
$x^{8} - x^{7} - 9 x^{6} + 10 x^{5} + 15 x^{4} - 10 x^{3} - 9 x^{2} + x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 29^{2}\cdot 1709$ |
$3$ |
$13.1576175741$ |
$497.80016070708535$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$46.6119375927$ |
8.8.1000118125.1 |
$x^{8} - 3 x^{7} - 4 x^{6} + 14 x^{5} + 5 x^{4} - 19 x^{3} - x^{2} + 7 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 31\cdot 41\cdot 1259$ |
$4$ |
$13.3354112142$ |
$2828.594173790224$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$7$ |
$49.7759598432$ |
8.8.1024000000.1 |
$x^{8} - 8 x^{6} + 19 x^{4} - 12 x^{2} + 1$ |
$8$ |
[8,0] |
$2^{16}\cdot 5^{6}$ |
$2$ |
$13.3748060995$ |
$13.37480609952844$ |
|
✓ |
✓ |
$C_4\times C_2$ (as 8T2) |
trivial |
$2$ |
$7$ |
$55.5791824785$ |
8.8.1032588125.1 |
$x^{8} - 9 x^{6} - 2 x^{5} + 23 x^{4} + 9 x^{3} - 17 x^{2} - 9 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 1652141$ |
$2$ |
$13.3887764654$ |
$2874.14422045937$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$7$ |
$54.3817514238$ |
8.8.1064390625.1 |
$x^{8} - 13 x^{6} + 44 x^{4} - 17 x^{2} + 1$ |
$8$ |
[8,0] |
$3^{4}\cdot 5^{6}\cdot 29^{2}$ |
$3$ |
$13.4396398271$ |
$31.187971562966524$ |
|
|
? |
$C_2^2:C_4$ (as 8T10) |
trivial |
$2$ |
$7$ |
$49.355192321$ |
8.8.1077044573.1 |
$x^{8} - x^{7} - 8 x^{6} + 8 x^{5} + 16 x^{4} - 17 x^{3} - 2 x^{2} + 5 x - 1$ |
$8$ |
[8,0] |
$11\cdot 97913143$ |
$2$ |
$13.459508764$ |
$32818.357256267416$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$54.1448931786$ |
8.8.1095205625.1 |
$x^{8} - 3 x^{7} - 5 x^{6} + 18 x^{5} + 2 x^{4} - 23 x^{3} + 2 x^{2} + 8 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 349\cdot 5021$ |
$3$ |
$13.4876708258$ |
$2960.007601341591$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$7$ |
$59.084256483$ |
8.8.1098290293.1 |
$x^{8} - 3 x^{7} - 4 x^{6} + 16 x^{5} + x^{4} - 23 x^{3} + 7 x^{2} + 5 x - 1$ |
$8$ |
[8,0] |
$7523\cdot 145991$ |
$2$ |
$13.4924135208$ |
$33140.46307763366$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$63.9120234189$ |
8.8.1104338125.1 |
$x^{8} - 2 x^{7} - 8 x^{6} + 15 x^{5} + 17 x^{4} - 31 x^{3} - 9 x^{2} + 17 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 11\cdot 29^{2}\cdot 191$ |
$4$ |
$13.5016783686$ |
$551.9465553837617$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$51.3049671711$ |
8.8.1114390153.1 |
$x^{8} - 8 x^{6} - 2 x^{5} + 16 x^{4} + 3 x^{3} - 10 x^{2} + 1$ |
$8$ |
[8,0] |
$347\cdot 3211499$ |
$2$ |
$13.5169795965$ |
$33382.48272672361$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$52.8950121075$ |
8.8.1121463125.1 |
$x^{8} - 3 x^{7} - 4 x^{6} + 15 x^{5} + 2 x^{4} - 18 x^{3} + 5 x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 19\cdot 94439$ |
$3$ |
$13.5276738752$ |
$2995.280454314754$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$7$ |
$57.2610108044$ |
8.8.1136700613.1 |
$x^{8} - x^{7} - 7 x^{6} + 4 x^{5} + 14 x^{4} - 4 x^{3} - 8 x^{2} + x + 1$ |
$8$ |
[8,0] |
$89\cdot 1301\cdot 9817$ |
$3$ |
$13.550513767$ |
$33714.99092391988$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$51.2625172436$ |
8.8.1142440000.1 |
$x^{8} - 3 x^{7} - 5 x^{6} + 15 x^{5} + 8 x^{4} - 15 x^{3} - 5 x^{2} + 4 x + 1$ |
$8$ |
[8,0] |
$2^{6}\cdot 5^{4}\cdot 13^{4}$ |
$3$ |
$13.5590472788$ |
$22.80350850198276$ |
|
|
✓ |
$Q_8:C_2$ (as 8T11) |
trivial |
$2$ |
$7$ |
$59.9995786491$ |
8.8.1152784549.1 |
$x^{8} - 4 x^{7} - x^{6} + 15 x^{5} - 3 x^{4} - 16 x^{3} + 4 x^{2} + 4 x - 1$ |
$8$ |
[8,0] |
$7\cdot 19^{2}\cdot 43\cdot 103^{2}$ |
$4$ |
$13.5743336077$ |
$767.500488599193$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$7$ |
$56.0040678259$ |
8.8.1153988125.1 |
$x^{8} - 2 x^{7} - 7 x^{6} + 11 x^{5} + 12 x^{4} - 16 x^{3} - 5 x^{2} + 6 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 101^{2}\cdot 181$ |
$3$ |
$13.576104351$ |
$302.33259830855155$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$53.31986114$ |
8.8.1166547493.1 |
$x^{8} - x^{7} - 7 x^{6} + 6 x^{5} + 14 x^{4} - 9 x^{3} - 9 x^{2} + 3 x + 1$ |
$8$ |
[8,0] |
$1166547493$ |
$1$ |
$13.5944863533$ |
$34154.757984796204$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$57.6779331487$ |
8.8.1183423341.1 |
$x^{8} - x^{7} - 8 x^{6} + 9 x^{5} + 17 x^{4} - 20 x^{3} - 8 x^{2} + 10 x - 1$ |
$8$ |
[8,0] |
$3\cdot 19^{2}\cdot 103^{3}$ |
$3$ |
$13.6189152203$ |
$244.09863248010305$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$7$ |
$66.8194726815$ |
8.8.1202043125.1 |
$x^{8} - 3 x^{7} - 4 x^{6} + 16 x^{5} - 21 x^{3} + 9 x^{2} + 2 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 41\cdot 61\cdot 769$ |
$4$ |
$13.6455173546$ |
$3101.023218229751$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$7$ |
$55.7182141133$ |
8.8.1225026133.1 |
$x^{8} - 3 x^{7} - 4 x^{6} + 18 x^{5} - 6 x^{4} - 17 x^{3} + 9 x^{2} + 2 x - 1$ |
$8$ |
[8,0] |
$83\cdot 14759351$ |
$2$ |
$13.6778604721$ |
$35000.37332658039$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$56.1212063278$ |
8.8.1243893125.1 |
$x^{8} - x^{7} - 8 x^{6} + 3 x^{5} + 18 x^{4} - x^{3} - 12 x^{2} - 2 x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 1990229$ |
$2$ |
$13.7040168337$ |
$3154.5435485978$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$7$ |
$54.6552881572$ |
8.8.1255718125.1 |
$x^{8} - 2 x^{7} - 8 x^{6} + 19 x^{5} + 10 x^{4} - 41 x^{3} + 13 x^{2} + 10 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 29^{2}\cdot 2389$ |
$3$ |
$13.7202340629$ |
$588.5618064400713$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$54.2354730586$ |
8.8.1261609229.1 |
$x^{8} - 2 x^{7} - 6 x^{6} + 12 x^{5} + 9 x^{4} - 19 x^{3} - x^{2} + 6 x - 1$ |
$8$ |
[8,0] |
$397\cdot 3177857$ |
$2$ |
$13.7282635234$ |
$35519.138911296825$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$61.0590696819$ |
8.8.1292203125.1 |
$x^{8} - 4 x^{7} - x^{6} + 17 x^{5} - 6 x^{4} - 21 x^{3} + 6 x^{2} + 8 x + 1$ |
$8$ |
[8,0] |
$3^{4}\cdot 5^{6}\cdot 1021$ |
$3$ |
$13.7694422325$ |
$185.05507578936948$ |
|
|
✓ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$2$ |
$7$ |
$53.9529294588$ |
8.8.1299600812.1 |
$x^{8} - 2 x^{7} - 6 x^{6} + 10 x^{5} + 12 x^{4} - 13 x^{3} - 8 x^{2} + 3 x + 1$ |
$8$ |
[8,0] |
$2^{2}\cdot 324900203$ |
$2$ |
$13.7792711639$ |
$36049.97658806452$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$91.1201780925$ |
8.8.1317743125.1 |
$x^{8} - x^{7} - 8 x^{6} + 7 x^{5} + 19 x^{4} - 14 x^{3} - 12 x^{2} + 8 x - 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 2108389$ |
$2$ |
$13.8031702571$ |
$3246.83615231813$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$7$ |
$58.4588266704$ |
8.8.1318279381.1 |
$x^{8} - x^{7} - 7 x^{6} + 5 x^{5} + 14 x^{4} - 6 x^{3} - 9 x^{2} + x + 1$ |
$8$ |
[8,0] |
$1318279381$ |
$1$ |
$13.8038722827$ |
$36308.117288011505$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$58.5595232418$ |
8.8.1326417388.1 |
$x^{8} - 2 x^{7} - 6 x^{6} + 10 x^{5} + 12 x^{4} - 13 x^{3} - 9 x^{2} + 4 x + 2$ |
$8$ |
[8,0] |
$2^{2}\cdot 43\cdot 2777^{2}$ |
$3$ |
$13.8144953894$ |
$691.1179349430892$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$7$ |
$92.233039402$ |
8.8.1348097653.1 |
$x^{8} - 2 x^{7} - 6 x^{6} + 11 x^{5} + 11 x^{4} - 17 x^{3} - 6 x^{2} + 6 x + 1$ |
$8$ |
[8,0] |
$7^{2}\cdot 27512197$ |
$2$ |
$13.8425203$ |
$13877.513430006113$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$72.5535341153$ |
8.8.1358954496.1 |
$x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 1$ |
$8$ |
[8,0] |
$2^{24}\cdot 3^{4}$ |
$2$ |
$13.8564064606$ |
$13.856406460551018$ |
|
✓ |
✓ |
$C_4\times C_2$ (as 8T2) |
trivial |
$2$ |
$7$ |
$93.4711768963$ |
8.8.1359341129.1 |
$x^{8} - 8 x^{6} - x^{5} + 18 x^{4} + 2 x^{3} - 12 x^{2} - x + 2$ |
$8$ |
[8,0] |
$1359341129$ |
$1$ |
$13.856899181$ |
$36869.24367274165$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$93.7187751003$ |
8.8.1377663125.1 |
$x^{8} - 12 x^{6} + 33 x^{4} - 5 x^{3} - 22 x^{2} + 5 x + 1$ |
$8$ |
[8,0] |
$5^{4}\cdot 29^{2}\cdot 2621$ |
$3$ |
$13.880109084$ |
$616.4778990361293$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$7$ |
$56.5417886705$ |
8.8.1381875749.1 |
$x^{8} - 3 x^{7} - 4 x^{6} + 14 x^{5} + 4 x^{4} - 18 x^{3} + x^{2} + 5 x - 1$ |
$8$ |
[8,0] |
$1381875749$ |
$1$ |
$13.885407332$ |
$37173.58940161684$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$7$ |
$74.1276222327$ |