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 |
8.2.4286875.1 |
$x^{8} - x^{7} + x^{5} - 2 x^{4} - x^{3} + 2 x^{2} + 2 x - 1$ |
$8$ |
[2,3] |
$-\,5^{4}\cdot 19^{3}$ |
$2$ |
$6.74555367758$ |
$9.746794344808963$ |
|
|
? |
$D_{8}$ (as 8T6) |
trivial |
$2$ |
$4$ |
$0.831403828897$ |
8.2.4296211.1 |
$x^{8} - x^{7} + 3 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 3 x + 1$ |
$8$ |
[2,3] |
$-\,199\cdot 21589$ |
$2$ |
$6.74738824726$ |
$2072.7303249578804$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.83312070768$ |
8.2.4297507.1 |
$x^{8} - x^{7} - 2 x^{6} + x^{5} + x^{3} + 2 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,2011\cdot 2137$ |
$2$ |
$6.74764264181$ |
$2073.0429325028463$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.83318984365$ |
8.2.4364587.1 |
$x^{8} - x^{7} + 2 x^{6} - 3 x^{5} + 3 x^{4} - 4 x^{3} + 2 x^{2} - 2 x + 1$ |
$8$ |
[2,3] |
$-\,29\cdot 150503$ |
$2$ |
$6.76071913838$ |
$2089.159400333062$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8412077944090119$ |
8.2.4386467.1 |
$x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} - 2 x^{3} + 2 x^{2} + x - 1$ |
$8$ |
[2,3] |
$-\,41\cdot 83\cdot 1289$ |
$3$ |
$6.76494637444$ |
$2094.3894098280766$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8443442011032247$ |
8.2.4421387.1 |
$x^{8} - 2 x^{7} + 2 x^{6} + x^{5} - 2 x^{4} + x^{3} + x^{2} - 1$ |
$8$ |
[2,3] |
$-\,1321\cdot 3347$ |
$2$ |
$6.77165488361$ |
$2102.7094426001895$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.848612129114202$ |
8.2.4423907.1 |
$x^{8} - x^{7} - 2 x^{6} + 3 x^{5} + 2 x^{4} - 2 x^{3} - 2 x^{2} + x + 1$ |
$8$ |
[2,3] |
$-\,4423907$ |
$1$ |
$6.772137207270576$ |
$2103.308584112184$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8486556559484179$ |
8.2.4456891.1 |
$x^{8} - x^{7} - x^{6} + 2 x^{5} - 2 x^{3} + x - 1$ |
$8$ |
[2,3] |
$-\,4456891$ |
$1$ |
$6.778428222351358$ |
$2111.1350027887843$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8533890882180042$ |
8.2.4461875.1 |
$x^{8} - x^{7} + x^{6} + 2 x^{5} - 2 x^{4} + 2 x^{2} - x - 1$ |
$8$ |
[2,3] |
$-\,5^{4}\cdot 11^{2}\cdot 59$ |
$3$ |
$6.7793752718$ |
$56.96490147450446$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$4$ |
$0.853816779689$ |
8.2.4505651.1 |
$x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + 5 x^{4} - 3 x^{3} + x^{2} + x - 1$ |
$8$ |
[2,3] |
$-\,4505651$ |
$1$ |
$6.787653961428009$ |
$2122.6518791360963$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8590899748258677$ |
8.2.4542739.1 |
$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 4 x^{4} - x^{3} - x^{2} + x - 1$ |
$8$ |
[2,3] |
$-\,4542739$ |
$1$ |
$6.794612960385022$ |
$2131.370216550846$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8640907466005752$ |
8.2.4570091.1 |
$x^{8} - x^{6} - x^{5} + x^{4} - x^{3} - 2 x^{2} + x + 1$ |
$8$ |
[2,3] |
$-\,1249\cdot 3659$ |
$2$ |
$6.799713367076283$ |
$2137.7771165395143$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8691071409989591$ |
8.2.4570723.1 |
$x^{8} - 2 x^{7} + 3 x^{6} - 3 x^{5} + 2 x^{3} - 4 x^{2} + 3 x - 1$ |
$8$ |
[2,3] |
$-\,4570723$ |
$1$ |
$6.7998309019$ |
$2137.9249285229826$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8669605427357915$ |
8.2.4584491.1 |
$x^{8} - x^{7} + 2 x^{6} - 2 x^{5} + 4 x^{4} - 5 x^{3} + 4 x^{2} - 3 x + 1$ |
$8$ |
[2,3] |
$-\,19\cdot 101\cdot 2389$ |
$3$ |
$6.80238785315$ |
$2141.1424520568453$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8688179378958599$ |
8.2.4596992.1 |
$x^{8} - 2 x^{7} + 3 x^{6} - 4 x^{5} + 4 x^{4} - 4 x^{3} + 2 x^{2} - 2 x + 1$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 17957$ |
$2$ |
$6.804703687569884$ |
$491.52785359466174$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8711682249043199$ |
8.2.4603987.1 |
$x^{8} - x^{7} - 2 x^{6} + x^{5} + 2 x^{4} - x - 1$ |
$8$ |
[2,3] |
$-\,4603987$ |
$1$ |
$6.8059971215378265$ |
$2145.690331804662$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8718212555217068$ |
8.2.4614499.1 |
$x^{8} - x^{7} + x^{5} + x^{4} - 2 x^{3} + 2 x - 1$ |
$8$ |
[2,3] |
$-\,4614499$ |
$1$ |
$6.807937648205655$ |
$2148.1384964661847$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8729503709676886$ |
8.2.4616192.1 |
$x^{8} - 2 x^{7} + 3 x^{6} - 4 x^{5} + 3 x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1$ |
$8$ |
[2,3] |
$-\,2^{12}\cdot 7^{2}\cdot 23$ |
$3$ |
$6.80824981614$ |
$35.888716889852724$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$4$ |
$0.873136825319$ |
8.2.4623371.1 |
$x^{8} + x^{6} - x^{3} - x^{2} - 1$ |
$8$ |
[2,3] |
$-\,17\cdot 31^{2}\cdot 283$ |
$3$ |
$6.809572421507061$ |
$684.4769184408666$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.873839750719525$ |
8.2.4648192.1 |
$x^{8} - 2 x^{7} + 2 x^{6} - 2 x^{4} + 2 x^{3} - x^{2} + 1$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 67\cdot 271$ |
$3$ |
$6.814131452560941$ |
$494.2575233501126$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8773171765831781$ |
8.2.4663051.1 |
$x^{8} - x^{6} - 2 x^{5} + x^{4} + x^{2} + 2 x - 1$ |
$8$ |
[2,3] |
$-\,31\cdot 359\cdot 419$ |
$3$ |
$6.816850516631215$ |
$2159.409873090331$ |
|
|
? |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8785601974574553$ |
8.2.4690927.1 |
$x^{8} - x^{7} - x^{6} + 2 x^{5} + x^{4} - 2 x^{3} + 2 x - 1$ |
$8$ |
[2,3] |
$-\,443\cdot 10589$ |
$2$ |
$6.8219311864100645$ |
$2165.854796610336$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8833361607728416$ |
8.2.4711123.1 |
$x^{8} - x^{7} - x^{5} + 2 x^{3} + x - 1$ |
$8$ |
[2,3] |
$-\,43\cdot 331^{2}$ |
$2$ |
$6.82559562487$ |
$119.30213744941874$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$4$ |
$0.884280803625$ |
8.2.4725251.1 |
$x^{8} - 4 x^{6} - 2 x^{5} + 7 x^{4} + 5 x^{3} - 3 x^{2} - 4 x - 1$ |
$8$ |
[2,3] |
$-\,59\cdot 283^{2}$ |
$2$ |
$6.82815090024$ |
$129.2168719633779$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$4$ |
$0.886962638902$ |
8.2.4761667.1 |
$x^{8} - x^{7} + x^{6} + x^{5} - 3 x^{4} + 4 x^{3} - 4 x^{2} + 3 x - 1$ |
$8$ |
[2,3] |
$-\,23\cdot 207029$ |
$2$ |
$6.834706625519568$ |
$2182.1244235835866$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.892409675427044$ |
8.2.4775363.1 |
$x^{8} - 2 x^{7} - x^{6} + 4 x^{5} - 2 x^{4} - 3 x^{3} + 3 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,1931\cdot 2473$ |
$2$ |
$6.837160875217895$ |
$2185.2603963830034$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.892761210808111$ |
8.2.4785667.1 |
$x^{8} - x^{7} + 2 x^{6} - x^{5} + x^{3} - 2 x^{2} + 2 x - 1$ |
$8$ |
[2,3] |
$-\,29\cdot 59\cdot 2797$ |
$3$ |
$6.839003240119339$ |
$2187.616739742133$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8954120433168544$ |
8.2.4809907.1 |
$x^{8} - 2 x^{7} + 3 x^{6} - 5 x^{5} + 5 x^{4} - 5 x^{3} + 4 x^{2} - 3 x + 1$ |
$8$ |
[2,3] |
$-\,19\cdot 253153$ |
$2$ |
$6.843323725534318$ |
$2193.15001766865$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.8973516802448191$ |
8.2.4858379.1 |
$x^{8} + 3 x^{6} - x^{5} + 2 x^{4} - 3 x^{3} - 2 x + 1$ |
$8$ |
[2,3] |
$-\,17^{2}\cdot 16811$ |
$2$ |
$6.851906434395191$ |
$534.5904974838218$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9045989163533711$ |
8.2.4931267.1 |
$x^{8} - 2 x^{7} + x^{6} + x^{5} - 3 x^{4} + 4 x^{3} - 4 x^{2} + 2 x - 1$ |
$8$ |
[2,3] |
$-\,11\cdot 67\cdot 6691$ |
$3$ |
$6.864672373301241$ |
$2220.6456268391858$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9159002625287757$ |
8.2.4960000.1 |
$x^{8} - 2 x^{7} + 4 x^{5} - 8 x^{4} + 10 x^{3} - 7 x^{2} + 4 x - 1$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 5^{4}\cdot 31$ |
$3$ |
$6.86965947012$ |
$35.21363372331802$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$4$ |
$0.915024622331$ |
8.2.5040467.1 |
$x^{8} - 3 x^{7} + 3 x^{6} - 4 x^{4} + 4 x^{3} - x + 1$ |
$8$ |
[2,3] |
$-\,5040467$ |
$1$ |
$6.883492554051752$ |
$2245.098438821781$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9288588955969742$ |
8.2.5040547.1 |
$x^{8} - 3 x^{7} + 4 x^{6} - 4 x^{5} + 2 x^{4} - x^{3} + x^{2} + 1$ |
$8$ |
[2,3] |
$-\,37\cdot 59\cdot 2309$ |
$3$ |
$6.883506210414857$ |
$2245.116255341803$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9268647190390614$ |
8.2.5103467.1 |
$x^{8} - x^{7} - x^{6} + x^{5} + x^{4} + x^{3} - 2 x^{2} + 1$ |
$8$ |
[2,3] |
$-\,5103467$ |
$1$ |
$6.894188661972919$ |
$2259.0854344180966$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9341515370278446$ |
8.2.5107019.1 |
$x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} - x^{4} + 4 x^{3} - 3 x^{2} + 2 x - 1$ |
$8$ |
[2,3] |
$-\,5107019$ |
$1$ |
$6.894788271629269$ |
$2259.871456521366$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9337002743186916$ |
8.2.5118587.1 |
$x^{8} - x^{7} - x^{6} + x^{5} + x^{2} - x - 1$ |
$8$ |
[2,3] |
$-\,29\cdot 176503$ |
$2$ |
$6.896738528281753$ |
$2262.429446413744$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9358964080809821$ |
8.2.5149367.1 |
$x^{8} - x^{6} - x^{5} - x^{3} + x^{2} + x - 1$ |
$8$ |
[2,3] |
$-\,47\cdot 331^{2}$ |
$2$ |
$6.90190902798$ |
$124.72770341828635$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$4$ |
$0.937944317141$ |
8.2.5155867.1 |
$x^{8} - x^{7} - 2 x^{6} + x^{5} + 3 x^{4} - x^{3} - 3 x^{2} + 1$ |
$8$ |
[2,3] |
$-\,449\cdot 11483$ |
$2$ |
$6.902997454301547$ |
$2270.653430182598$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9405039275650298$ |
8.2.5165819.1 |
$x^{8} - 2 x^{6} - x^{5} + 2 x^{4} + 3 x^{3} - x - 1$ |
$8$ |
[2,3] |
$-\,641\cdot 8059$ |
$2$ |
$6.904661594549214$ |
$2272.843813375657$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9414451308962976$ |
8.2.5204491.1 |
$x^{8} - x^{7} + x^{6} - 2 x^{4} + x^{3} - x^{2} - x + 1$ |
$8$ |
[2,3] |
$-\,5204491$ |
$1$ |
$6.911101682109057$ |
$2281.335354567583$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9486003676975986$ |
8.2.5233147.1 |
$x^{8} - 2 x^{7} + 2 x^{5} + x^{4} - x^{3} - 2 x^{2} + x + 1$ |
$8$ |
[2,3] |
$-\,5233147$ |
$1$ |
$6.915846841129293$ |
$2287.607265244627$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9515125685677428$ |
8.2.5272027.1 |
$x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} - 3 x^{3} + 2 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,317\cdot 16631$ |
$2$ |
$6.922248776687624$ |
$2296.08950173986$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9542437305912765$ |
8.2.5286727.1 |
$x^{8} - x^{7} - x^{6} + 2 x^{5} - x^{4} - 2 x^{3} + 2 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,5286727$ |
$1$ |
$6.924658503091751$ |
$2299.2883681695953$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9546924661447783$ |
8.2.5293867.1 |
$x^{8} - x^{7} + 2 x^{5} - x^{4} - x^{3} + x^{2} + x - 1$ |
$8$ |
[2,3] |
$-\,227\cdot 23321$ |
$2$ |
$6.925826826912146$ |
$2300.840498600457$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9584670756759093$ |
8.2.5344939.1 |
$x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + 4 x^{4} - x^{3} - x^{2} + 2 x - 1$ |
$8$ |
[2,3] |
$-\,521\cdot 10259$ |
$2$ |
$6.934143805490464$ |
$2311.912411835708$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9628254972480232$ |
8.2.5346947.1 |
$x^{8} - x^{6} - x^{5} + x^{4} + 2 x^{3} - 2 x - 1$ |
$8$ |
[2,3] |
$-\,839\cdot 6373$ |
$2$ |
$6.934469381534787$ |
$2312.3466435636333$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.9655694198061011$ |
8.2.5359051.1 |
$x^{8} - x^{7} + x^{6} - 2 x^{4} + 4 x^{3} - 4 x^{2} + 3 x - 1$ |
$8$ |
[2,3] |
$-\,5359051$ |
$1$ |
$6.936429654555921$ |
$2314.9624187014356$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$4$ |
$0.966607895416933$ |
8.2.5365963.1 |
$x^{8} - x^{7} + x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{3} + 3 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,67\cdot 283^{2}$ |
$2$ |
$6.93754733307$ |
$137.69894698217558$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$4$ |
$0.964152352733$ |
8.2.5365963.2 |
$x^{8} - 2 x^{5} + x^{4} - x^{3} - 2 x^{2} + 3 x + 1$ |
$8$ |
[2,3] |
$-\,67\cdot 283^{2}$ |
$2$ |
$6.93754733307$ |
$137.69894698217558$ |
|
|
? |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$4$ |
$0.96720132041$ |
8.2.5369375.1 |
$x^{8} - 2 x^{7} + 2 x^{6} - 4 x^{3} + 5 x^{2} - 4 x + 1$ |
$8$ |
[2,3] |
$-\,5^{4}\cdot 11^{2}\cdot 71$ |
$3$ |
$6.93809859314$ |
$62.489999199871974$ |
|
|
? |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$4$ |
$0.964343847115$ |