| 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 |
| 8.0.4227136.1 |
$x^{8} + x^{4} + x^{2} + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 257^{2}$ |
$2$ |
$6.73373122385$ |
$45.34313619501854$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$0.76270871296$ |
| 8.0.8248384.1 |
$x^{8} - x^{7} - x^{6} + 3 x^{5} + 2 x^{4} - 3 x^{3} - x^{2} + x + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 359^{2}$ |
$2$ |
$7.32059041525$ |
$53.59104402789705$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$1.60799968131$ |
| 8.0.9684544.1 |
$x^{8} - x^{7} - x^{5} + 2 x^{4} + x^{3} + x + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 389^{2}$ |
$2$ |
$7.46895593265$ |
$44.27682411416183$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$1.96178080331$ |
| 8.0.11999296.1 |
$x^{8} - x^{7} + x^{6} + x^{5} - 2 x^{4} + x^{3} + x^{2} - x + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 433^{2}$ |
$2$ |
$7.67175050938$ |
$58.855755878248644$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$2.14094108769$ |
| 8.0.12730624.1 |
$x^{8} + x^{6} + 2 x^{4} - 3 x^{2} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 223^{2}$ |
$2$ |
$7.72869575622$ |
$59.732738092272314$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$2.14229042288$ |
| 8.0.13424896.1 |
$x^{8} + x^{2} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 229^{2}$ |
$2$ |
$7.78016605232$ |
$50.90028729061861$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$2.05971422249$ |
| 8.0.13424896.2 |
$x^{8} - x^{2} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 229^{2}$ |
$2$ |
$7.78016605232$ |
$50.90028729061861$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$1.32912589603$ |
| 8.0.16908544.1 |
$x^{8} + x^{4} - x^{2} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 257^{2}$ |
$2$ |
$8.00780108192$ |
$53.92238017965345$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$2.56120019223$ |
| 8.0.20502784.1 |
$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 283^{2}$ |
$2$ |
$8.20307353161$ |
$56.58426906139804$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$1.85480050158$ |
| 8.4.20502784.1 |
$x^{8} - x^{6} + 4 x^{4} - 4 x^{2} + 1$ |
$8$ |
[4,2] |
$2^{8}\cdot 283^{2}$ |
$2$ |
$8.20307353161$ |
$56.58426906139804$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$5$ |
$2.7869643821$ |
| 8.0.20866624.1 |
$x^{8} - x^{7} + x^{5} + x^{3} - x + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 571^{2}$ |
$2$ |
$8.22113015314$ |
$53.64382236530028$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$3.21452612329$ |
| 8.0.28047616.1 |
$x^{8} - x^{6} - 2 x^{4} + 2 x^{2} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 331^{2}$ |
$2$ |
$8.53074566463$ |
$61.19507752396393$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$1.84817637649$ |
| 8.4.28047616.1 |
$x^{8} + x^{6} - 2 x^{4} - 2 x^{2} + 1$ |
$8$ |
[4,2] |
$2^{8}\cdot 331^{2}$ |
$2$ |
$8.53074566463$ |
$61.19507752396393$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$5$ |
$3.4110845539$ |
| 8.0.31091776.1 |
$x^{8} + 3 x^{4} - x^{2} + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 17^{2}\cdot 41^{2}$ |
$3$ |
$8.64133200441$ |
$74.67261881037788$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$2.81041870785$ |
| 8.0.32171584.1 |
$x^{8} - x^{7} - x^{5} + x^{3} + x + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 709^{2}$ |
$2$ |
$8.67828793801$ |
$59.77571494764291$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$4.28480353183$ |
| 8.0.32993536.1 |
$x^{8} - 2 x^{7} + 4 x^{6} - 8 x^{5} + 6 x^{4} - 4 x^{3} + 3 x^{2} + 2 x + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 359^{2}$ |
$2$ |
$8.70569820784$ |
$53.59104402789705$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$3.59988093776$ |
| 8.4.32993536.1 |
$x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 14 x^{4} - 6 x^{3} - 3 x^{2} + 4 x - 1$ |
$8$ |
[4,2] |
$2^{8}\cdot 359^{2}$ |
$2$ |
$8.70569820784$ |
$53.59104402789705$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$5$ |
$3.76255994435$ |
| 8.0.36096064.1 |
$x^{8} - x^{6} + 3 x^{4} + 4 x^{2} + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 751^{2}$ |
$2$ |
$8.80404960801$ |
$77.5112895003044$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$6.37976462001$ |
| 8.0.37063744.1 |
$x^{8} - x^{6} + x^{4} + 2 x^{2} + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 761^{2}$ |
$2$ |
$8.83321214579$ |
$78.02563681252464$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$3.04460569117$ |
| 8.0.38738176.1 |
$x^{8} - 2 x^{5} + x^{4} + 2 x^{3} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 389^{2}$ |
$2$ |
$8.88213553675$ |
$44.27682411416183$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$4.55420648809$ |
| 8.4.38738176.1 |
$x^{8} - 2 x^{7} + 2 x^{5} - 3 x^{4} + 2 x^{3} - 2 x + 1$ |
$8$ |
[4,2] |
$2^{8}\cdot 389^{2}$ |
$2$ |
$8.88213553675$ |
$44.27682411416183$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$5$ |
$4.64473891267$ |
| 8.0.39740416.1 |
$x^{8} + 4 x^{4} + 2 x^{2} + 1$ |
$8$ |
[0,4] |
$2^{10}\cdot 197^{2}$ |
$2$ |
$8.91054055404$ |
$63.01802241635832$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$5.08041031252$ |
| 8.0.40347904.1 |
$x^{8} - x^{7} + 3 x^{6} + x^{5} + x^{3} + 3 x^{2} - x + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 397^{2}$ |
$2$ |
$8.92745402568$ |
$79.6994353806851$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$4.41235740333$ |
| 8.0.43138624.1 |
$x^{8} - x^{7} + 3 x^{6} - 3 x^{5} + 2 x^{4} - 3 x^{3} + 3 x^{2} - x + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 821^{2}$ |
$2$ |
$9.00239958886$ |
$81.04319835741924$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$6.74012286784$ |
| 8.0.43178041.1 |
$x^{8} + 2 x^{6} - 3 x^{5} + 4 x^{4} - 3 x^{3} + 2 x^{2} + 1$ |
$8$ |
[0,4] |
$6571^{2}$ |
$1$ |
$9.00342739694$ |
$81.0617048920142$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$7.88150646357$ |
| 8.0.43178041.2 |
$x^{8} - x^{6} - 2 x^{4} + 4 x^{2} + 1$ |
$8$ |
[0,4] |
$6571^{2}$ |
$1$ |
$9.00342739694$ |
$81.0617048920142$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$4.33067473633$ |
| 8.0.43178041.3 |
$x^{8} - 3 x^{7} + 2 x^{6} + 3 x^{5} - 5 x^{4} + 3 x^{3} + 2 x^{2} - 3 x + 1$ |
$8$ |
[0,4] |
$6571^{2}$ |
$1$ |
$9.00342739694$ |
$81.0617048920142$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$2.36888754709$ |
| 8.0.47375689.1 |
$x^{8} - 2 x^{7} + 2 x^{6} - x^{5} + 4 x^{4} - x^{3} + 2 x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$6883^{2}$ |
$1$ |
$9.10844923948$ |
$82.96384754819415$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$8.3469103632$ |
| 8.0.47375689.2 |
$x^{8} - x^{7} + 2 x^{6} - 5 x^{5} + 7 x^{4} - 5 x^{3} + 2 x^{2} - x + 1$ |
$8$ |
[0,4] |
$6883^{2}$ |
$1$ |
$9.10844923948$ |
$82.96384754819415$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$3.54773229851$ |
| 8.0.47375689.3 |
$x^{8} - 3 x^{7} + 2 x^{6} + x^{5} - x^{4} + x^{3} + 2 x^{2} - 3 x + 1$ |
$8$ |
[0,4] |
$6883^{2}$ |
$1$ |
$9.10844923948$ |
$82.96384754819415$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$4.49794354851$ |
| 8.0.47997184.1 |
$x^{8} - x^{6} - 2 x^{5} + 5 x^{4} - 2 x^{3} - x^{2} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 433^{2}$ |
$2$ |
$9.12330029029$ |
$58.855755878248644$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$4.97983318852$ |
| 8.0.47997184.2 |
$x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + x^{4} + 2 x^{3} - x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 433^{2}$ |
$2$ |
$9.12330029029$ |
$58.855755878248644$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$3.73150696778$ |
| 8.0.56806369.1 |
$x^{8} + x^{6} - x^{5} + x^{3} + x^{2} + 1$ |
$8$ |
[0,4] |
$7537^{2}$ |
$1$ |
$9.31750487855$ |
$86.81589716175259$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$3.9777971709$ |
| 8.4.56806369.1 |
$x^{8} - x^{7} - x^{6} + x^{5} - x^{4} + x^{3} - x^{2} - x + 1$ |
$8$ |
[4,2] |
$7537^{2}$ |
$1$ |
$9.31750487855$ |
$86.81589716175259$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$5$ |
$6.02155161965$ |
| 8.0.56806369.2 |
$x^{8} - x^{7} + x^{6} - x^{5} + x^{4} + x^{3} + x^{2} + x + 1$ |
$8$ |
[0,4] |
$7537^{2}$ |
$1$ |
$9.31750487855$ |
$86.81589716175259$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$4.29919395147$ |
| 8.4.57957769.1 |
$x^{8} - x^{7} + x^{6} + 2 x^{5} - 11 x^{4} + 14 x^{3} - 14 x^{2} + 8 x - 1$ |
$8$ |
[4,2] |
$23^{2}\cdot 331^{2}$ |
$2$ |
$9.34090504862$ |
$87.25250712730265$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$5$ |
$5.29366927269$ |
| 8.0.57957769.1 |
$x^{8} - x^{7} + 5 x^{6} - 6 x^{5} + 7 x^{4} - 10 x^{3} + 6 x^{2} - 2 x + 5$ |
$8$ |
[0,4] |
$23^{2}\cdot 331^{2}$ |
$2$ |
$9.34090504862$ |
$87.25250712730265$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$3.71191684842$ |
| 8.0.61716736.1 |
$x^{8} + 2 x^{6} + 2 x^{4} + 3 x^{2} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 491^{2}$ |
$2$ |
$9.41456739445$ |
$74.53207948931465$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$5.35844867244$ |
| 8.4.61716736.1 |
$x^{8} - 2 x^{6} + 2 x^{4} - 3 x^{2} + 1$ |
$8$ |
[4,2] |
$2^{8}\cdot 491^{2}$ |
$2$ |
$9.41456739445$ |
$74.53207948931465$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$5$ |
$5.6107375293$ |
| 8.0.62094400.1 |
$x^{8} + 2 x^{6} + 5 x^{4} + 3 x^{2} + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 5^{2}\cdot 197^{2}$ |
$3$ |
$9.42174952356$ |
$88.76936408468859$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$6.09095444314$ |
| 8.0.63616576.1 |
$x^{8} - x^{7} + 4 x^{6} - x^{5} + 4 x^{4} + x^{3} + 4 x^{2} + x + 1$ |
$8$ |
[0,4] |
$2^{6}\cdot 997^{2}$ |
$2$ |
$9.45031503428$ |
$70.88416711072195$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$6.8959452589$ |
| 8.0.64979721.1 |
$x^{8} - 3 x^{7} + 8 x^{6} - 14 x^{5} + 21 x^{4} - 19 x^{3} + 13 x^{2} - 5 x + 2$ |
$8$ |
[0,4] |
$3^{2}\cdot 2687^{2}$ |
$2$ |
$9.47539296816$ |
$89.78307190111062$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$9.6202367411$ |
| 8.4.64979721.1 |
$x^{8} - 2 x^{7} - 3 x^{6} + 2 x^{5} + 5 x^{4} + 3 x^{3} - 2 x^{2} - 4 x - 1$ |
$8$ |
[4,2] |
$3^{2}\cdot 2687^{2}$ |
$2$ |
$9.47539296816$ |
$89.78307190111062$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$5$ |
$6.52522980503$ |
| 8.0.65108761.1 |
$x^{8} + 2 x^{6} - x^{5} + 2 x^{4} + x^{3} + 2 x^{2} + 1$ |
$8$ |
[0,4] |
$8069^{2}$ |
$1$ |
$9.4777430162$ |
$89.8276126811795$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$10.1234587533$ |
| 8.4.65108761.1 |
$x^{8} - x^{7} - 2 x^{6} + x^{5} + x^{4} + x^{3} - 2 x^{2} - x + 1$ |
$8$ |
[4,2] |
$8069^{2}$ |
$1$ |
$9.4777430162$ |
$89.8276126811795$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$5$ |
$5.60016476966$ |
| 8.0.65108761.2 |
$x^{8} - x^{7} + 2 x^{6} - x^{5} + 3 x^{4} + x^{3} + 2 x^{2} + x + 1$ |
$8$ |
[0,4] |
$8069^{2}$ |
$1$ |
$9.4777430162$ |
$89.8276126811795$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$5.58731072006$ |
| 8.4.68425984.1 |
$x^{8} + x^{6} - 2 x^{4} - 3 x^{2} + 1$ |
$8$ |
[4,2] |
$2^{8}\cdot 11^{2}\cdot 47^{2}$ |
$3$ |
$9.53679904408$ |
$90.95053600721658$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$5$ |
$9.28949206428$ |
| 8.0.71639296.1 |
$x^{8} - x^{7} + x^{6} + x^{5} + 4 x^{4} + x^{3} + x^{2} - x + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 23^{4}$ |
$2$ |
$9.59166304663$ |
$32.350317596360256$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$5.43700947665$ |
| 8.0.74097664.1 |
$x^{8} - 2 x^{6} + 4 x^{4} + 4 x^{2} + 1$ |
$8$ |
[0,4] |
$2^{10}\cdot 269^{2}$ |
$2$ |
$9.63220161946$ |
$73.63898559015824$ |
|
|
? |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$6.72516234294$ |
| 8.0.81144064.1 |
$x^{8} - 2 x^{6} + 3 x^{2} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 563^{2}$ |
$2$ |
$9.74220119591$ |
$67.11184694225007$ |
|
|
✓ |
$C_2^3:S_4$ (as 8T39) |
trivial |
$2$ |
$3$ |
$6.53043479045$ |
