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 |
8.0.1257728.1 |
$x^{8} - 2 x^{7} + 4 x^{5} - 4 x^{4} + 3 x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 17^{3}$ |
$2$ |
$5.78693149384$ |
$16.74428805718538$ |
|
|
✓ |
$C_4\wr C_2$ (as 8T17) |
trivial |
$4$ |
$3$ |
$0.618886617067$ |
8.0.1265625.1 |
$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 5^{6}$ |
$2$ |
$5.79146092644$ |
$5.791460926441346$ |
✓ |
✓ |
✓ |
$C_4\times C_2$ (as 8T2) |
trivial |
$30$ |
$3$ |
$4.6618207773$ |
8.0.1282789.1 |
$x^{8} - x^{7} + 2 x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{3} + 3 x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$1103\cdot 1163$ |
$2$ |
$5.80122090621$ |
$1132.6027547202946$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$3$ |
$0.313539838403$ |
8.0.1327833.1 |
$x^{8} - x^{7} + x^{6} - 2 x^{5} + 3 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 13^{2}\cdot 97$ |
$3$ |
$5.82630119728$ |
$61.506097258727124$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$6$ |
$3$ |
$0.963083858701$ |
8.0.1342413.1 |
$x^{8} - x^{7} + x^{6} + x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 16573$ |
$2$ |
$5.83425985742$ |
$222.97757734803739$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$6$ |
$3$ |
$0.970727427757$ |
8.0.1361513.1 |
$x^{8} - x^{7} + 2 x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{3} + 2 x^{2} - x + 1$ |
$8$ |
[0,4] |
$17\cdot 283^{2}$ |
$2$ |
$5.84457214701$ |
$69.36137253543934$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$3$ |
$0.32641205382$ |
8.0.1385533.1 |
$x^{8} - 2 x^{7} + 3 x^{6} - 3 x^{5} + x^{4} + 1$ |
$8$ |
[0,4] |
$29\cdot 47777$ |
$2$ |
$5.85736259192$ |
$1177.0866578124144$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$3$ |
$0.331112529152$ |
8.0.1424293.1 |
$x^{8} - 2 x^{7} + 2 x^{6} - x^{5} + x^{4} - x^{3} + 2 x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$13\cdot 331^{2}$ |
$2$ |
$5.87759852004$ |
$65.5972560401729$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$3$ |
$0.336709575883$ |
8.0.1474013.1 |
$x^{8} - x^{7} + x^{6} - x^{4} + x^{3} - x^{2} + 1$ |
$8$ |
[0,4] |
$617\cdot 2389$ |
$2$ |
$5.90286246256$ |
$1214.0893706807585$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$3$ |
$0.345105135911$ |
8.0.1492101.1 |
$x^{8} - x^{7} + x^{6} - 3 x^{5} + x^{4} - 2 x^{3} + 3 x^{2} + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 13^{2}\cdot 109$ |
$3$ |
$5.91186866834$ |
$65.19969325081216$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$6$ |
$3$ |
$1.04325557006$ |
8.0.1513728.1 |
$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 3 x^{4} - 2 x^{3} + x^{2} + 2 x + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 3^{4}\cdot 73$ |
$3$ |
$5.92251242534$ |
$29.597297173897484$ |
|
|
✓ |
$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) |
trivial |
$12$ |
$3$ |
$2.10694405423$ |
8.0.1520789.1 |
$x^{8} - x^{7} - x^{6} + x^{4} - x^{2} + x + 1$ |
$8$ |
[0,4] |
$29\cdot 229^{2}$ |
$2$ |
$5.9259586988$ |
$81.49233092751734$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$3$ |
$0.353845256559$ |
8.0.1578125.1 |
$x^{8} - 2 x^{7} + 3 x^{5} - x^{4} - 3 x^{3} + 2 x + 1$ |
$8$ |
[0,4] |
$5^{6}\cdot 101$ |
$2$ |
$5.95343580388$ |
$33.60378443921746$ |
|
|
✓ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$10$ |
$3$ |
$1.81195906066$ |
8.0.1590773.1 |
$x^{8} - x^{7} + 2 x^{5} - 2 x^{4} + x^{3} + x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$179\cdot 8887$ |
$2$ |
$5.95937927669$ |
$1261.2584984847476$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$3$ |
$0.363609073488$ |
8.0.1601613.1 |
$x^{8} - 2 x^{6} - 3 x^{5} + 3 x^{4} + 3 x^{3} - 2 x^{2} + 1$ |
$8$ |
[0,4] |
$3^{6}\cdot 13^{3}$ |
$2$ |
$5.9644403303$ |
$15.606246247497747$ |
|
|
✓ |
$C_4\wr C_2$ (as 8T17) |
trivial |
$6$ |
$3$ |
$1.10095849199$ |
8.0.1740113.1 |
$x^{8} - 2 x^{7} + 3 x^{6} - 4 x^{5} + 4 x^{4} - 4 x^{3} + 3 x^{2} - x + 1$ |
$8$ |
[0,4] |
$1740113$ |
$1$ |
$6.0265974216$ |
$1319.1334276713633$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$3$ |
$0.38876666774366264$ |
8.0.1750329.1 |
$x^{8} - 3 x^{7} + 4 x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{3} + 4 x^{2} - 3 x + 1$ |
$8$ |
[0,4] |
$3^{6}\cdot 7^{4}$ |
$2$ |
$6.03100878452$ |
$6.03100878451909$ |
|
✓ |
? |
$D_4$ (as 8T4) |
trivial |
$6$ |
$3$ |
$1.1656993933$ |
8.0.1763584.1 |
$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 3 x^{4} + 2 x^{3} + x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 83^{2}$ |
$2$ |
$6.03669895858$ |
$18.2208671582886$ |
|
|
✓ |
$S_4\times C_2$ (as 8T24) |
trivial |
$4$ |
$3$ |
$0.781195313952$ |
8.0.1797309.1 |
$x^{8} - 3 x^{7} + 6 x^{6} - 8 x^{5} + 10 x^{4} - 9 x^{3} + 7 x^{2} - 4 x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 22189$ |
$2$ |
$6.05100962346$ |
$258.005813887982$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$6$ |
$3$ |
$1.20014144807$ |
8.0.1820637.1 |
$x^{8} - x^{7} + x^{5} - x^{4} - 2 x^{3} + 2 x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 7\cdot 13^{2}\cdot 19$ |
$4$ |
$6.06077163648$ |
$72.02083032012335$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$6$ |
$3$ |
$1.21081359453$ |
8.0.1867553.1 |
$x^{8} - x^{7} + x^{5} - 2 x^{4} + x^{3} + x^{2} - x + 1$ |
$8$ |
[0,4] |
$1867553$ |
$1$ |
$6.08007752654$ |
$1366.5844284199934$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$3$ |
$0.40885972396034426$ |
8.0.1890625.1 |
$x^{8} - 3 x^{7} + 6 x^{6} - 9 x^{5} + 9 x^{4} - 7 x^{3} + 4 x^{2} - x + 1$ |
$8$ |
[0,4] |
$5^{6}\cdot 11^{2}$ |
$2$ |
$6.08941642815$ |
$11.08980336897316$ |
|
|
✓ |
$C_2^2:C_4$ (as 8T10) |
trivial |
$10$ |
$3$ |
$2.04169489066$ |
8.0.2067245.1 |
$x^{8} - x^{7} + x^{6} - x^{4} + x^{2} - x + 1$ |
$8$ |
[0,4] |
$5\cdot 643^{2}$ |
$2$ |
$6.15777751924$ |
$56.70097000933934$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$3$ |
$0.453534943932$ |
8.0.2105893.1 |
$x^{8} - x^{7} + 2 x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$29\cdot 72617$ |
$2$ |
$6.17205144936$ |
$1451.1695283460165$ |
|
|
✓ |
$S_8$ (as 8T50) |
trivial |
$2$ |
$3$ |
$0.44069755405256955$ |
8.0.2118069.1 |
$x^{8} - x^{7} - x^{3} + 2 x^{2} - x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 79\cdot 331$ |
$3$ |
$6.17650095624$ |
$280.083915996617$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$6$ |
$3$ |
$1.33864883478$ |
8.0.2149173.1 |
$x^{8} - x^{7} - 2 x^{6} + 2 x^{5} + x^{3} + x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 13^{2}\cdot 157$ |
$3$ |
$6.18776657236$ |
$78.24960063795854$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$6$ |
$3$ |
$1.34468819349$ |
8.0.2178981.1 |
$x^{8} - x^{7} - x^{6} + 2 x^{5} + x^{4} - 2 x^{3} - x^{2} + x + 1$ |
$8$ |
[0,4] |
$3^{6}\cdot 7^{2}\cdot 61$ |
$3$ |
$6.19842970625$ |
$47.1036844046804$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$6$ |
$3$ |
$1.34805397566$ |
8.0.2193125.1 |
$x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + 5 x^{4} - 5 x^{3} + 4 x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$5^{4}\cdot 11^{2}\cdot 29$ |
$3$ |
$6.20344481555$ |
$39.93745109543172$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$2$ |
$3$ |
$0.454073872008$ |
8.0.2217213.1 |
$x^{8} - x^{7} - x^{6} - x^{5} + 3 x^{4} - x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 31\cdot 883$ |
$3$ |
$6.21192104402$ |
$286.5641289484781$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$6$ |
$3$ |
$1.37910404568$ |
8.0.2313441.1 |
$x^{8} - x^{7} + 2 x^{6} + 3 x^{5} - x^{4} + 3 x^{3} + 2 x^{2} - x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 13^{4}$ |
$2$ |
$6.2449979984$ |
$6.244997998398398$ |
|
✓ |
? |
$D_4$ (as 8T4) |
trivial |
$6$ |
$3$ |
$1.41187738145$ |
8.0.2314413.1 |
$x^{8} - x^{6} - x^{5} - x^{3} + 2 x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 28573$ |
$2$ |
$6.24532592023$ |
$292.77807294946115$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$6$ |
$3$ |
$1.42029894182$ |
8.0.2477709.1 |
$x^{8} - x^{7} + 3 x^{5} - x^{3} + 2 x^{2} + 2 x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 13^{2}\cdot 181$ |
$3$ |
$6.2987777876$ |
$84.01785524517987$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$6$ |
$3$ |
$1.51551812359$ |
8.0.2560000.1 |
$x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 2 x^{4} + 2 x^{3} + 2 x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$2^{12}\cdot 5^{4}$ |
$2$ |
$6.32455532034$ |
$6.324555320336759$ |
|
✓ |
? |
$D_4$ (as 8T4) |
trivial |
$4$ |
$3$ |
$1.0839823354$ |
8.0.2671805.1 |
$x^{8} - x^{5} + x^{4} - x^{3} + 1$ |
$8$ |
[0,4] |
$5\cdot 17^{2}\cdot 43^{2}$ |
$3$ |
$6.35844025272$ |
$60.45659600076736$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$3$ |
$0.542256876594$ |
8.0.2708009.1 |
$x^{8} - x^{7} - x^{6} + x^{5} + x^{4} - x^{3} - x^{2} + x + 1$ |
$8$ |
[0,4] |
$41\cdot 257^{2}$ |
$2$ |
$6.36914685919$ |
$102.6498904042279$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$3$ |
$0.563263410972$ |
8.0.2712933.1 |
$x^{8} - 3 x^{7} + 3 x^{6} - x^{5} + x^{4} - 2 x^{2} + x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 33493$ |
$2$ |
$6.37059334437$ |
$316.9842267369151$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$6$ |
$3$ |
$1.57666375612$ |
8.0.2824713.1 |
$x^{8} - 3 x^{7} + 2 x^{6} + 3 x^{5} - 4 x^{4} + x^{2} + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 43\cdot 811$ |
$3$ |
$6.40282734558$ |
$323.4486048818266$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$6$ |
$3$ |
$1.7400253268$ |
8.0.2828125.1 |
$x^{8} - x^{7} - x^{6} - 2 x^{5} + 4 x^{4} + x^{3} + x^{2} - 3 x + 1$ |
$8$ |
[0,4] |
$5^{6}\cdot 181$ |
$2$ |
$6.40379359023$ |
$44.98490324139099$ |
|
|
✓ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$10$ |
$3$ |
$2.66425859071$ |
8.0.2844928.1 |
$x^{8} - 2 x^{7} + 2 x^{6} - 2 x^{4} + 2 x^{3} + x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 11113$ |
$2$ |
$6.40853720479$ |
$210.83642948978243$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$4$ |
$3$ |
$1.11974748455$ |
8.0.2963293.1 |
$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 7 x^{4} - 6 x^{3} + 5 x^{2} - 3 x + 1$ |
$8$ |
[0,4] |
$37\cdot 283^{2}$ |
$2$ |
$6.44127475268$ |
$102.32790430767162$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$3$ |
$0.545793348928$ |
8.0.2970513.1 |
$x^{8} - 2 x^{7} + 2 x^{6} - 2 x^{5} + 3 x^{4} - 3 x^{2} + x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 7\cdot 13^{2}\cdot 31$ |
$4$ |
$6.44323441823$ |
$91.9945650568554$ |
|
|
✓ |
$C_2 \wr C_2\wr C_2$ (as 8T35) |
trivial |
$6$ |
$3$ |
$1.66725651269$ |
8.0.2985984.1 |
$x^{8} - 2 x^{7} + 2 x^{6} - 2 x^{5} + 7 x^{4} - 10 x^{3} + 8 x^{2} - 4 x + 1$ |
$8$ |
[0,4] |
$2^{12}\cdot 3^{6}$ |
$2$ |
$6.44741959094$ |
$6.4474195909412515$ |
|
✓ |
? |
$D_4$ (as 8T4) |
trivial |
$12$ |
$3$ |
$3.64163813098$ |
8.0.3057993.1 |
$x^{8} - 2 x^{7} + 3 x^{6} - 2 x^{5} + x^{3} + 2 x^{2} - 3 x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 19\cdot 1987$ |
$3$ |
$6.46665305207$ |
$336.5397450524975$ |
|
|
? |
$S_4\wr C_2$ (as 8T47) |
trivial |
$6$ |
$3$ |
$1.83028307754$ |
8.0.3134053.1 |
$x^{8} - x^{7} + x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} + x^{2} - x + 1$ |
$8$ |
[0,4] |
$13\cdot 491^{2}$ |
$2$ |
$6.48654286035$ |
$79.89367934949547$ |
|
|
✓ |
$C_2 \wr S_4$ (as 8T44) |
trivial |
$2$ |
$3$ |
$0.570147165225$ |
8.0.3167829.1 |
$x^{8} - 3 x^{7} + 4 x^{6} - 4 x^{4} + 3 x^{3} + 2 x^{2} - 3 x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 7\cdot 37\cdot 151$ |
$4$ |
$6.49524019945$ |
$342.53029063135426$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$6$ |
$3$ |
$1.77871765724$ |
8.0.3198933.1 |
$x^{8} - x^{7} - x^{6} + 4 x^{4} - x^{3} - 3 x^{2} + x + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 73\cdot 541$ |
$3$ |
$6.50317802496$ |
$344.20778608276714$ |
|
|
✓ |
$S_4\wr C_2$ (as 8T47) |
trivial |
$6$ |
$3$ |
$2.04684730841$ |
8.0.3203125.1 |
$x^{8} - x^{7} + 2 x^{6} + 2 x^{5} + 3 x^{3} + 2 x^{2} + x + 1$ |
$8$ |
[0,4] |
$5^{7}\cdot 41$ |
$2$ |
$6.50424266534$ |
$26.181268353554465$ |
|
|
✓ |
$((C_8 : C_2):C_2):C_2$ (as 8T27) |
trivial |
$10$ |
$3$ |
$3.04388544877$ |
8.0.3207681.1 |
$x^{8} - 3 x^{7} + 4 x^{6} - 3 x^{5} + 2 x^{4} - x^{2} + 1$ |
$8$ |
[0,4] |
$3^{4}\cdot 199^{2}$ |
$2$ |
$6.50539836897$ |
$24.43358344574123$ |
|
|
✓ |
$S_4\times C_2$ (as 8T24) |
trivial |
$6$ |
$3$ |
$1.89392987116$ |
8.0.3211264.1 |
$x^{8} - 4 x^{7} + 8 x^{6} - 8 x^{5} + 7 x^{4} - 8 x^{3} + 8 x^{2} - 4 x + 1$ |
$8$ |
[0,4] |
$2^{16}\cdot 7^{2}$ |
$2$ |
$6.50630624679$ |
$10.583005244258363$ |
|
|
? |
$D_4\times C_2$ (as 8T9) |
trivial |
$8$ |
$3$ |
$2.5180422861$ |
8.0.3240000.1 |
$x^{8} - 3 x^{7} + 3 x^{6} + x^{5} - 2 x^{4} - 3 x^{3} + 7 x^{2} - 4 x + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 3^{4}\cdot 5^{4}$ |
$3$ |
$6.51355562433$ |
$10.954451150103322$ |
|
|
✓ |
$Q_8:C_2$ (as 8T11) |
trivial |
$6$ |
$3$ |
$1.89812936734$ |