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 |
8.6.65106259.1 |
$x^{8} - 5 x^{6} - x^{5} + 7 x^{4} + 4 x^{3} - 4 x^{2} - 2 x + 1$ |
$8$ |
[6,1] |
$-\,89\cdot 731531$ |
$2$ |
$9.47769748909$ |
$8068.844960711539$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$7.13506329205$ |
8.6.68494627.1 |
$x^{8} - 2 x^{7} - 3 x^{6} + 10 x^{5} - 2 x^{4} - 11 x^{3} + 5 x^{2} + 2 x - 1$ |
$8$ |
[6,1] |
$-\,811\cdot 84457$ |
$2$ |
$9.53799440023$ |
$8276.14807745729$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$7.38088275852$ |
8.6.68856875.1 |
$x^{8} - 2 x^{7} - 2 x^{6} + x^{5} + x^{4} + 5 x^{3} - x^{2} - 3 x + 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 29^{2}\cdot 131$ |
$3$ |
$9.54428531564$ |
$137.82234942127494$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$6$ |
$7.41473348387$ |
8.6.69367411.1 |
$x^{8} - x^{6} - 3 x^{5} - 3 x^{4} + 6 x^{3} + 4 x^{2} - 2 x - 1$ |
$8$ |
[6,1] |
$-\,69367411$ |
$1$ |
$9.55310245976$ |
$8328.71004417851$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$7.43035213338$ |
8.6.73061875.1 |
$x^{8} - 4 x^{6} - 3 x^{4} - 5 x^{3} + 4 x^{2} + 5 x + 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 29^{2}\cdot 139$ |
$3$ |
$9.61526707297$ |
$141.96830632222108$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$6$ |
$7.66924336082$ |
8.6.74671875.1 |
$x^{8} - 4 x^{7} + 5 x^{6} - x^{5} - 6 x^{4} + 9 x^{3} - 4 x + 1$ |
$8$ |
[6,1] |
$-\,3^{4}\cdot 5^{6}\cdot 59$ |
$3$ |
$9.64150058785$ |
$44.48505546908213$ |
|
|
✓ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$2$ |
$6$ |
$7.76874863093$ |
8.6.74906875.1 |
$x^{8} - 3 x^{6} - 3 x^{5} + 3 x^{4} + 7 x^{3} - 2 x^{2} - 3 x + 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 119851$ |
$2$ |
$9.64528822399$ |
$774.1156244386235$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$6$ |
$7.80639541536$ |
8.6.75272867.1 |
$x^{8} - 2 x^{7} - 3 x^{6} + 9 x^{5} - x^{4} - 10 x^{3} + 3 x^{2} + 3 x - 1$ |
$8$ |
[6,1] |
$-\,31\cdot 2428157$ |
$2$ |
$9.6511664817$ |
$8675.993718301092$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$7.82545816415$ |
8.6.83260171.1 |
$x^{8} - 2 x^{7} - 3 x^{6} + 4 x^{5} + 3 x^{4} + x^{3} - x^{2} - 3 x - 1$ |
$8$ |
[6,1] |
$-\,47\cdot 1771493$ |
$2$ |
$9.77360228296$ |
$9124.701145791023$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$8.34345597927$ |
8.6.84356875.1 |
$x^{8} - 3 x^{6} - x^{5} + 4 x^{3} + 3 x^{2} - 2 x - 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 71\cdot 1901$ |
$3$ |
$9.78960255478$ |
$821.49558732838$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$6$ |
$8.3932670464$ |
8.6.86606875.1 |
$x^{8} - x^{7} - 3 x^{6} + 2 x^{5} - x^{4} + 6 x^{3} + 3 x^{2} - 7 x - 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 138571$ |
$2$ |
$9.82186693421$ |
$832.3791203532198$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$6$ |
$8.61675807033$ |
8.6.93594547.1 |
$x^{8} - x^{7} - 6 x^{6} + 4 x^{5} + 11 x^{4} - 3 x^{3} - 7 x^{2} + x + 1$ |
$8$ |
[6,1] |
$-\,1117\cdot 83791$ |
$2$ |
$9.91759383729$ |
$9674.42747660036$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$8.97005663928$ |
8.6.94086875.1 |
$x^{8} - 2 x^{7} + x^{6} + 4 x^{5} - 12 x^{4} + 3 x^{3} + 9 x^{2} - 6 x + 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 29^{2}\cdot 179$ |
$3$ |
$9.92409997105$ |
$161.1055554597668$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$6$ |
$8.98090007444$ |
8.6.94156147.1 |
$x^{8} - 5 x^{6} - x^{5} + 7 x^{4} + 4 x^{3} - 3 x^{2} - 3 x + 1$ |
$8$ |
[6,1] |
$-\,94156147$ |
$1$ |
$9.92501301141$ |
$9703.409040126053$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.0269732907$ |
8.6.94856875.1 |
$x^{8} - x^{7} - x^{6} + 3 x^{5} - 5 x^{4} - 2 x^{3} + 6 x^{2} + x - 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 151771$ |
$2$ |
$9.93421606822$ |
$871.1228386398786$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$6$ |
$9.1297954509$ |
8.6.95029219.1 |
$x^{8} - x^{7} - 3 x^{6} + 3 x^{5} - x^{4} - 2 x^{3} + 4 x^{2} + x - 1$ |
$8$ |
[6,1] |
$-\,1291\cdot 73609$ |
$2$ |
$9.93647044244$ |
$9748.293132646351$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.10258386395$ |
8.6.96530491.1 |
$x^{8} - 2 x^{7} - 3 x^{6} + 7 x^{5} - 6 x^{3} + 4 x^{2} + x - 1$ |
$8$ |
[6,1] |
$-\,96530491$ |
$1$ |
$9.95595819643$ |
$9824.99318065921$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.17845856742$ |
8.6.96598171.1 |
$x^{8} - 3 x^{7} + x^{6} + 4 x^{5} - 7 x^{4} + 5 x^{3} + 5 x^{2} - 6 x + 1$ |
$8$ |
[6,1] |
$-\,53\cdot 1229\cdot 1483$ |
$3$ |
$9.95683047606$ |
$9828.436854352782$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.13243508284$ |
8.6.97161811.1 |
$x^{8} - 5 x^{6} - x^{5} + 6 x^{4} + 3 x^{3} - 2 x - 1$ |
$8$ |
[6,1] |
$-\,43\cdot 521\cdot 4337$ |
$3$ |
$9.96407413503$ |
$9857.06908771568$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.26321506754$ |
8.6.98070859.1 |
$x^{8} - 2 x^{7} - x^{6} + 3 x^{5} - 5 x^{4} + 10 x^{3} - 4 x^{2} - 2 x + 1$ |
$8$ |
[6,1] |
$-\,61\cdot 263\cdot 6113$ |
$3$ |
$9.97567972486$ |
$9903.073209867733$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.27226479393$ |
8.6.98151947.1 |
$x^{8} - 4 x^{6} - x^{5} + 3 x^{4} + 3 x^{3} + 2 x^{2} - 2 x - 1$ |
$8$ |
[6,1] |
$-\,7297\cdot 13451$ |
$2$ |
$9.97671037691$ |
$9907.166446567859$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.2719804595$ |
8.6.98915419.1 |
$x^{8} - 4 x^{6} - x^{5} + 2 x^{4} + x^{3} + 3 x^{2} - 1$ |
$8$ |
[6,1] |
$-\,151\cdot 655069$ |
$2$ |
$9.98637796823$ |
$9945.623107679075$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.3162069083$ |
8.6.103405923.1 |
$x^{8} - x^{7} + x^{5} - 8 x^{4} + x^{3} + 8 x^{2} - 2 x - 1$ |
$8$ |
[6,1] |
$-\,3^{3}\cdot 19^{2}\cdot 103^{2}$ |
$3$ |
$10.0419528277$ |
$76.62245101796209$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$6$ |
$9.68622718301$ |
8.6.104081867.1 |
$x^{8} - 3 x^{7} - x^{6} + 10 x^{5} - 5 x^{4} - 10 x^{3} + 7 x^{2} + 3 x - 1$ |
$8$ |
[6,1] |
$-\,19\cdot 73\cdot 75041$ |
$3$ |
$10.0501347389$ |
$10202.052097494896$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.59842021162$ |
8.6.104429827.1 |
$x^{8} - x^{7} - 2 x^{6} + x^{5} - 2 x^{4} + 5 x^{3} + 2 x^{2} - 4 x + 1$ |
$8$ |
[6,1] |
$-\,17\cdot 199\cdot 30869$ |
$3$ |
$10.0543284817$ |
$10219.091300110787$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.62967810582$ |
8.6.106143931.1 |
$x^{8} - 4 x^{7} + 3 x^{6} + 5 x^{5} - 8 x^{4} + 2 x^{3} + 5 x^{2} - 2 x - 1$ |
$8$ |
[6,1] |
$-\,3659\cdot 29009$ |
$2$ |
$10.0748107278$ |
$10302.617677076054$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.048074566$ |
8.6.106899371.1 |
$x^{8} - 3 x^{7} + 3 x^{5} + 3 x^{4} - 6 x^{2} + 1$ |
$8$ |
[6,1] |
$-\,67\cdot 1595513$ |
$2$ |
$10.0837459083$ |
$10339.215202325562$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.081255108$ |
8.6.107555299.1 |
$x^{8} - 3 x^{7} - x^{6} + 9 x^{5} - 5 x^{4} - 5 x^{3} + 6 x^{2} - 1$ |
$8$ |
[6,1] |
$-\,1153\cdot 93283$ |
$2$ |
$10.0914593814$ |
$10370.887088383519$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.1197503358$ |
8.6.107781875.1 |
$x^{8} - 2 x^{7} - x^{6} + 4 x^{5} - 5 x^{4} - 3 x^{3} + 5 x^{2} + x - 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 331\cdot 521$ |
$3$ |
$10.0941142692$ |
$928.5768681159358$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$6$ |
$9.85579681337$ |
8.6.108493747.1 |
$x^{8} - x^{7} - 5 x^{6} + 5 x^{5} + 7 x^{4} - 8 x^{3} - 3 x^{2} + 4 x + 1$ |
$8$ |
[6,1] |
$-\,108493747$ |
$1$ |
$10.1024239211$ |
$10416.033170070072$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.96053466815$ |
8.6.108714131.1 |
$x^{8} - x^{7} - 4 x^{6} + 5 x^{5} + 3 x^{4} - 9 x^{3} + x^{2} + 4 x - 1$ |
$8$ |
[6,1] |
$-\,103\cdot 971\cdot 1087$ |
$3$ |
$10.1049867837$ |
$10426.606878558336$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.0232921526$ |
8.6.109020787.1 |
$x^{8} - 5 x^{6} - 3 x^{5} + 8 x^{4} + 9 x^{3} - 4 x^{2} - 6 x - 1$ |
$8$ |
[6,1] |
$-\,109020787$ |
$1$ |
$10.1085453568$ |
$10441.30197820176$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.9328123712$ |
8.6.109380619.1 |
$x^{8} - 2 x^{7} - 2 x^{6} + 2 x^{5} + x^{4} + 5 x^{3} + x^{2} - 4 x - 1$ |
$8$ |
[6,1] |
$-\,109380619$ |
$1$ |
$10.1127098567$ |
$10458.518967808013$ |
|
|
? |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$9.90248579451$ |
8.6.110906875.1 |
$x^{8} - 2 x^{7} - 4 x^{6} + 9 x^{5} - 9 x^{3} + 4 x^{2} + 3 x - 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 29^{2}\cdot 211$ |
$3$ |
$10.1302417263$ |
$174.91426471274434$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$6$ |
$9.94997062$ |
8.6.111219379.1 |
$x^{8} - 2 x^{7} - 2 x^{6} + 7 x^{5} - 3 x^{4} - 7 x^{3} + 4 x^{2} + 2 x - 1$ |
$8$ |
[6,1] |
$-\,29\cdot 389\cdot 9859$ |
$3$ |
$10.1338053527$ |
$10546.059880353421$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.034952344$ |
8.6.112129379.1 |
$x^{8} - x^{7} - 4 x^{6} + 2 x^{5} + 4 x^{4} + 3 x^{3} - x^{2} - 4 x - 1$ |
$8$ |
[6,1] |
$-\,1427\cdot 78577$ |
$2$ |
$10.1441328247$ |
$10589.116063203764$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.1802489898$ |
8.6.114655339.1 |
$x^{8} - 4 x^{7} + 3 x^{6} + 4 x^{5} - 5 x^{4} + 3 x^{3} - x^{2} - 3 x + 1$ |
$8$ |
[6,1] |
$-\,233\cdot 492083$ |
$2$ |
$10.1724200532$ |
$10707.723334117294$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.1164006344$ |
8.6.118871243.1 |
$x^{8} - 2 x^{6} - 2 x^{5} - 3 x^{4} + 5 x^{3} + 5 x^{2} - 2 x - 1$ |
$8$ |
[6,1] |
$-\,101\cdot 1176943$ |
$2$ |
$10.2184400303$ |
$10902.808949990824$ |
|
|
? |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.6488116417$ |
8.6.119649419.1 |
$x^{8} - x^{7} - 4 x^{6} + 6 x^{5} + 6 x^{4} - 9 x^{3} - 4 x^{2} + 3 x + 1$ |
$8$ |
[6,1] |
$-\,119649419$ |
$1$ |
$10.2267779$ |
$10938.437685519812$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.444006294$ |
8.6.121587131.1 |
$x^{8} - 2 x^{7} - x^{6} + 7 x^{5} - 7 x^{4} - 5 x^{3} + 8 x^{2} + x - 1$ |
$8$ |
[6,1] |
$-\,23\cdot 271\cdot 19507$ |
$3$ |
$10.2473354061$ |
$11026.655476616652$ |
|
|
? |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.9088684379$ |
8.6.123531875.1 |
$x^{8} - 5 x^{6} - 3 x^{5} + 8 x^{4} + 10 x^{3} - 4 x^{2} - 7 x + 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 197651$ |
$2$ |
$10.2676812783$ |
$994.1101548621259$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$6$ |
$10.8492896616$ |
8.6.128782547.1 |
$x^{8} - 2 x^{7} + 5 x^{5} - 8 x^{4} - 3 x^{3} + 8 x^{2} - x - 1$ |
$8$ |
[6,1] |
$-\,157\cdot 820271$ |
$2$ |
$10.321245928$ |
$11348.23981946099$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$11.1025063692$ |
8.6.129062827.1 |
$x^{8} - 2 x^{7} - 4 x^{6} + 5 x^{5} + 8 x^{4} - 3 x^{3} - 6 x^{2} + x + 1$ |
$8$ |
[6,1] |
$-\,17\cdot 31\cdot 244901$ |
$3$ |
$10.3240511296$ |
$11360.58215937898$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$10.9881510415$ |
8.6.131931875.1 |
$x^{8} - x^{7} - 5 x^{6} + 5 x^{5} + 3 x^{4} - 3 x + 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 29^{2}\cdot 251$ |
$3$ |
$10.3524637111$ |
$190.77473627291428$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$6$ |
$11.2866336082$ |
8.6.132655867.1 |
$x^{8} - 2 x^{6} - 2 x^{5} - 3 x^{4} + 3 x^{3} + 4 x^{2} - x - 1$ |
$8$ |
[6,1] |
$-\,137\cdot 968291$ |
$2$ |
$10.3595480187$ |
$11517.632873121109$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$11.5180770626$ |
8.6.132761875.1 |
$x^{8} - x^{7} - 5 x^{6} + 2 x^{5} + 11 x^{4} - x^{3} - 9 x^{2} + 1$ |
$8$ |
[6,1] |
$-\,5^{4}\cdot 212419$ |
$2$ |
$10.3605824728$ |
$1030.57993382367$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$2$ |
$6$ |
$12.0204012295$ |
8.6.134560000.1 |
$x^{8} - 3 x^{6} + 4 x^{2} - 1$ |
$8$ |
[6,1] |
$-\,2^{8}\cdot 5^{4}\cdot 29^{2}$ |
$3$ |
$10.3780198565$ |
$44.16876366153594$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$6$ |
$11.4855032762$ |
8.6.134560000.2 |
$x^{8} - 6 x^{4} + 5 x^{2} - 1$ |
$8$ |
[6,1] |
$-\,2^{8}\cdot 5^{4}\cdot 29^{2}$ |
$3$ |
$10.3780198565$ |
$44.16876366153594$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$6$ |
$11.1552211117$ |
8.6.135123147.1 |
$x^{8} - 4 x^{6} - 4 x^{5} + 9 x^{4} + 5 x^{3} - 6 x^{2} - x + 1$ |
$8$ |
[6,1] |
$-\,3^{4}\cdot 257\cdot 6491$ |
$3$ |
$10.3834390726$ |
$2237.087615628856$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$11.2672389878$ |
8.6.136128151.1 |
$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 7 x^{4} + 2 x^{3} + 6 x^{2} - x - 1$ |
$8$ |
[6,1] |
$-\,97\cdot 1403383$ |
$2$ |
$10.3930614045$ |
$11667.396924764324$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$6$ |
$11.5512184196$ |