| 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 |
| 8.2.22665187.2 |
$x^{8} - x^{7} + x^{6} - 2 x^{5} + x^{4} - 9 x^{3} + 7 x^{2} - 6 x + 1$ |
$8$ |
[2,3] |
$-\,283^{3}$ |
$1$ |
$8.30653526758$ |
$16.822603841260722$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$2.50378244423$ |
| 8.2.22665187.3 |
$x^{8} - 3 x^{7} + 4 x^{6} - 8 x^{5} + 8 x^{4} - x^{3} + 2 x^{2} - 3 x - 1$ |
$8$ |
[2,3] |
$-\,283^{3}$ |
$1$ |
$8.30653526758$ |
$16.822603841260722$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$2.50378244423$ |
| 8.2.32019867.1 |
$x^{8} - 4 x^{7} + 7 x^{6} - 7 x^{5} + 7 x^{4} - 7 x^{3} + 2 x^{2} + x - 1$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 11^{4}$ |
$2$ |
$8.67316164265$ |
$17.819817582388065$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$3.73623245724$ |
| 8.2.32019867.2 |
$x^{8} - x^{7} + x^{6} - 2 x^{5} - x^{4} - 2 x^{3} + x^{2} - x + 1$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 11^{4}$ |
$2$ |
$8.67316164265$ |
$17.819817582388065$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$3.73623245724$ |
| 8.2.36264691.2 |
$x^{8} - 4 x^{7} + 5 x^{6} - x^{5} - 3 x^{4} + 3 x^{3} - x - 1$ |
$8$ |
[2,3] |
$-\,331^{3}$ |
$1$ |
$8.80918027485$ |
$18.193405398660254$ |
|
|
✓ |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$3.68334441406$ |
| 8.2.36264691.3 |
$x^{8} - 2 x^{7} + 3 x^{6} - 7 x^{4} + 17 x^{3} - 17 x^{2} + 11 x - 1$ |
$8$ |
[2,3] |
$-\,331^{3}$ |
$1$ |
$8.80918027485$ |
$18.193405398660254$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$3.68334441406$ |
| 8.2.81415168.2 |
$x^{8} - 4 x^{7} + 8 x^{6} - 8 x^{5} + 6 x^{3} - 2 x^{2} - 2$ |
$8$ |
[2,3] |
$-\,2^{10}\cdot 43^{3}$ |
$2$ |
$9.74626387324$ |
$16.52370966031933$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$8.25075445397$ |
| 8.2.81415168.3 |
$x^{8} - 4 x^{7} + 10 x^{6} - 14 x^{5} + 10 x^{4} + 2 x^{3} - 10 x^{2} + 8 x - 2$ |
$8$ |
[2,3] |
$-\,2^{10}\cdot 43^{3}$ |
$2$ |
$9.74626387324$ |
$16.52370966031933$ |
|
|
✓ |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$8.25075445397$ |
| 8.2.110716875.1 |
$x^{8} - 4 x^{7} + 5 x^{6} - x^{5} - 2 x^{4} + x^{3} + 4 x^{2} - 4 x - 1$ |
$8$ |
[2,3] |
$-\,3^{11}\cdot 5^{4}$ |
$2$ |
$10.128070772$ |
$21.91302343598798$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$9.22238270846$ |
| 8.2.110716875.2 |
$x^{8} - 4 x^{7} + 7 x^{6} - 7 x^{5} + 4 x^{4} - x^{3} - 4 x^{2} + 4 x - 1$ |
$8$ |
[2,3] |
$-\,3^{11}\cdot 5^{4}$ |
$2$ |
$10.128070772$ |
$21.91302343598798$ |
|
|
✓ |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$9.22238270846$ |
| 8.2.118370771.2 |
$x^{8} - 2 x^{7} - x^{5} + 7 x^{3} + x - 9$ |
$8$ |
[2,3] |
$-\,491^{3}$ |
$1$ |
$10.213052386$ |
$22.15851980616034$ |
|
|
✓ |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$10.0346697251$ |
| 8.2.118370771.3 |
$x^{8} - 3 x^{7} + 7 x^{6} - 10 x^{5} + 11 x^{4} - 5 x^{3} + 3 x^{2} - 2 x - 3$ |
$8$ |
[2,3] |
$-\,491^{3}$ |
$1$ |
$10.213052386$ |
$22.15851980616034$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$10.0346697251$ |
| 8.2.150730227.3 |
$x^{8} - 2 x^{7} + 2 x^{6} - x^{5} + 2 x^{4} + x^{3} - 5 x^{2} + 4 x - 1$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 41^{3}$ |
$2$ |
$10.526283055$ |
$23.06924557594881$ |
|
|
✓ |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$9.12401416235$ |
| 8.2.150730227.4 |
$x^{8} + 2 x^{6} - 2 x^{5} - 3 x^{4} - 7 x^{3} - 11 x^{2} - 6 x - 1$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 41^{3}$ |
$2$ |
$10.526283055$ |
$23.06924557594881$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$14.3057573471$ |
| 8.2.150730227.5 |
$x^{8} - 4 x^{7} + 7 x^{6} - 6 x^{5} - 3 x^{4} + 9 x^{3} - x^{2} - 5 x - 1$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 41^{3}$ |
$2$ |
$10.526283055$ |
$23.06924557594881$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$14.3057573471$ |
| 8.2.150730227.6 |
$x^{8} - x^{7} + x^{6} + 5 x^{5} - 11 x^{4} + 8 x^{3} - 7 x^{2} + 4 x - 1$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 41^{3}$ |
$2$ |
$10.526283055$ |
$23.06924557594881$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$9.12401416235$ |
| 8.2.178453547.1 |
$x^{8} - x^{7} - 2 x^{6} + 6 x^{5} - 11 x^{3} + 5 x^{2} + 8 x - 3$ |
$8$ |
[2,3] |
$-\,563^{3}$ |
$1$ |
$10.7507966688$ |
$23.727621035409346$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$12.347157199$ |
| 8.2.178453547.2 |
$x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 4 x^{4} + 12 x^{3} - 7 x^{2} + 2 x - 9$ |
$8$ |
[2,3] |
$-\,563^{3}$ |
$1$ |
$10.7507966688$ |
$23.727621035409346$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$12.347157199$ |
| 8.2.181398528.1 |
$x^{8} - 6 x^{4} + 4 x^{2} - 3$ |
$8$ |
[2,3] |
$-\,2^{10}\cdot 3^{11}$ |
$2$ |
$10.7728154462$ |
$18.88407114158967$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$16.6622023168$ |
| 8.2.181398528.2 |
$x^{8} - 2 x^{6} - 2 x^{5} - 2 x^{3} - 2 x^{2} + 1$ |
$8$ |
[2,3] |
$-\,2^{10}\cdot 3^{11}$ |
$2$ |
$10.7728154462$ |
$18.88407114158967$ |
|
|
✓ |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$16.6622023168$ |
| 8.2.182660427.1 |
$x^{8} - 4 x^{7} + 7 x^{6} - 7 x^{5} + 4 x^{4} - x^{3} + 4 x^{2} - 4 x + 1$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 17^{4}$ |
$2$ |
$10.7821546968$ |
$17.289418182735098$ |
|
|
✓ |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$9.96103143264$ |
| 8.2.182660427.2 |
$x^{8} - x^{7} + 4 x^{6} - 4 x^{5} - 2 x^{4} - 4 x^{3} - 5 x^{2} - x + 1$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 17^{4}$ |
$2$ |
$10.7821546968$ |
$17.289418182735098$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$9.96103143264$ |
| 8.2.265847707.1 |
$x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 7 x^{4} + 18 x^{3} - 12 x^{2} + 4 x - 1$ |
$8$ |
[2,3] |
$-\,643^{3}$ |
$1$ |
$11.3000174684$ |
$25.357444666211933$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$10.712578457$ |
| 8.2.265847707.2 |
$x^{8} - x^{7} - x^{6} + 5 x^{5} - 3 x^{4} + 2 x^{3} - 6 x^{2} + 9 x - 1$ |
$8$ |
[2,3] |
$-\,643^{3}$ |
$1$ |
$11.3000174684$ |
$25.357444666211933$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$10.712578457$ |
| 8.2.286557184.3 |
$x^{8} + 5 x^{6} + 6 x^{4} - 3 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,2^{10}\cdot 23^{4}$ |
$2$ |
$11.4064739398$ |
$22.875128945924832$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$19.2067057184$ |
| 8.2.286557184.4 |
$x^{8} - 5 x^{6} + 6 x^{4} + 3 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,2^{10}\cdot 23^{4}$ |
$2$ |
$11.4064739398$ |
$22.875128945924832$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$19.2067057184$ |
| 8.2.325660672.2 |
$x^{8} - 2 x^{7} + 4 x^{6} - 12 x^{5} + 16 x^{4} - 18 x^{3} + 10 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,2^{12}\cdot 43^{3}$ |
$2$ |
$11.5903263428$ |
$19.65011309429094$ |
|
|
✓ |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$16.7435125502$ |
| 8.2.325660672.3 |
$x^{8} - 2 x^{7} + 6 x^{6} - 12 x^{5} + 16 x^{4} - 10 x^{3} - 4 x^{2} + 4 x - 1$ |
$8$ |
[2,3] |
$-\,2^{12}\cdot 43^{3}$ |
$2$ |
$11.5903263428$ |
$19.65011309429094$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$16.7435125502$ |
| 8.2.423564751.1 |
$x^{8} - x^{7} - 2 x^{6} + 4 x^{5} - 9 x^{4} + 17 x^{3} - 21 x^{2} + 7 x - 4$ |
$8$ |
[2,3] |
$-\,751^{3}$ |
$1$ |
$11.977466818$ |
$27.40437921208944$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$37.2086722771$ |
| 8.2.423564751.2 |
$x^{8} - 2 x^{7} + 2 x^{6} - 4 x^{5} - 4 x^{4} + 20 x^{3} - 14 x^{2} + 13 x - 4$ |
$8$ |
[2,3] |
$-\,751^{3}$ |
$1$ |
$11.977466818$ |
$27.40437921208944$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$37.2086722771$ |
| 8.2.425329947.1 |
$x^{8} - x^{7} - 3 x^{6} + 2 x^{5} + 4 x^{4} + 3 x^{3} - 5 x^{2} - 7 x - 3$ |
$8$ |
[2,3] |
$-\,3^{11}\cdot 7^{4}$ |
$2$ |
$11.9836949472$ |
$27.423380759717027$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$26.3870692873$ |
| 8.2.425329947.2 |
$x^{8} - 6 x^{4} - x^{2} - 3$ |
$8$ |
[2,3] |
$-\,3^{11}\cdot 7^{4}$ |
$2$ |
$11.9836949472$ |
$27.423380759717027$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$26.3870692873$ |
| 8.2.725594112.1 |
$x^{8} - 6 x^{4} - 4 x^{2} - 3$ |
$8$ |
[2,3] |
$-\,2^{12}\cdot 3^{11}$ |
$2$ |
$12.8111087772$ |
$22.45707176179599$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$59.7703393538$ |
| 8.2.725594112.2 |
$x^{8} - 4 x^{6} + 12 x^{2} - 12$ |
$8$ |
[2,3] |
$-\,2^{12}\cdot 3^{11}$ |
$2$ |
$12.8111087772$ |
$22.45707176179599$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$59.7703393538$ |
| 8.2.941391011.1 |
$x^{8} + 2 x^{6} - 9 x^{4} + 6 x^{2} - 11$ |
$8$ |
[2,3] |
$-\,11^{3}\cdot 29^{4}$ |
$2$ |
$13.2349180308$ |
$31.30605480404366$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$21.4586885036$ |
| 8.2.941391011.2 |
$x^{8} - 2 x^{7} - 2 x^{6} + 17 x^{5} - 32 x^{4} + 31 x^{3} - 15 x^{2} + 4 x - 1$ |
$8$ |
[2,3] |
$-\,11^{3}\cdot 29^{4}$ |
$2$ |
$13.2349180308$ |
$31.30605480404366$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$21.4586885036$ |
| 8.2.999406512.1 |
$x^{8} - 4 x^{7} + 7 x^{6} - 7 x^{5} - 2 x^{4} + 11 x^{3} - 2 x^{2} - 4 x - 2$ |
$8$ |
[2,3] |
$-\,2^{4}\cdot 3^{7}\cdot 13^{4}$ |
$3$ |
$13.3342247785$ |
$22.950738422598956$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$85.8364832866$ |
| 8.2.999406512.2 |
$x^{8} - 4 x^{7} + x^{6} + 11 x^{5} - 11 x^{4} - x^{3} + 4 x^{2} - x - 2$ |
$8$ |
[2,3] |
$-\,2^{4}\cdot 3^{7}\cdot 13^{4}$ |
$3$ |
$13.3342247785$ |
$22.950738422598956$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$85.8364832866$ |
| 8.2.1280239375.1 |
$x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 18 x^{4} - 14 x^{3} + 4 x^{2} + x - 1$ |
$8$ |
[2,3] |
$-\,5^{4}\cdot 127^{3}$ |
$2$ |
$13.7534419117$ |
$32.95200640537238$ |
|
|
✓ |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$32.6933534109$ |
| 8.2.1280239375.2 |
$x^{8} - 2 x^{7} + 5 x^{5} - 6 x^{4} - 10 x^{3} + 21 x^{2} + 5 x - 13$ |
$8$ |
[2,3] |
$-\,5^{4}\cdot 127^{3}$ |
$2$ |
$13.7534419117$ |
$32.95200640537238$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$32.6933534109$ |
| 8.2.1327373299.2 |
$x^{8} - 3 x^{7} + 2 x^{6} - 3 x^{4} + 11 x^{3} - 13 x^{2} + 15 x - 11$ |
$8$ |
[2,3] |
$-\,7^{3}\cdot 157^{3}$ |
$2$ |
$13.8157394611$ |
$33.15116890850155$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$31.4318260199$ |
| 8.2.1327373299.3 |
$x^{8} - x^{7} - 2 x^{6} - 6 x^{5} + 10 x^{4} + 15 x^{3} - 6 x^{2} - 5 x - 7$ |
$8$ |
[2,3] |
$-\,7^{3}\cdot 157^{3}$ |
$2$ |
$13.8157394611$ |
$33.15116890850155$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$31.4318260199$ |
| 8.2.1327373299.4 |
$x^{8} - 6 x^{6} - 5 x^{5} + 19 x^{4} + 21 x^{3} - 18 x^{2} - 36 x - 5$ |
$8$ |
[2,3] |
$-\,7^{3}\cdot 157^{3}$ |
$2$ |
$13.8157394611$ |
$33.15116890850155$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$28.7258633658$ |
| 8.2.1327373299.5 |
$x^{8} - x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} - 9 x^{3} + 2 x^{2} - 3 x - 5$ |
$8$ |
[2,3] |
$-\,7^{3}\cdot 157^{3}$ |
$2$ |
$13.8157394611$ |
$33.15116890850155$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$28.7258633658$ |
| 8.2.1399680000.1 |
$x^{8} - 6 x^{4} - 12 x^{2} - 3$ |
$8$ |
[2,3] |
$-\,2^{10}\cdot 3^{7}\cdot 5^{4}$ |
$3$ |
$13.9076449382$ |
$19.267901981093516$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$73.165309216$ |
| 8.2.1399680000.2 |
$x^{8} - 6 x^{6} + 12 x^{4} - 6 x^{3} - 6 x^{2} + 18 x - 3$ |
$8$ |
[2,3] |
$-\,2^{10}\cdot 3^{7}\cdot 5^{4}$ |
$3$ |
$13.9076449382$ |
$19.267901981093516$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$73.165309216$ |
| 8.2.1626347584.1 |
$x^{8} - x^{7} - x^{6} - 4 x^{5} + 6 x^{4} - 4 x^{3} - x^{2} - x + 1$ |
$8$ |
[2,3] |
$-\,2^{6}\cdot 71^{4}$ |
$2$ |
$14.1710382773$ |
$34.292743267870684$ |
|
|
|
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$76.3102907518$ |
| 8.2.1626347584.2 |
$x^{8} - 8 x^{6} + 19 x^{4} - 16 x^{2} - 4$ |
$8$ |
[2,3] |
$-\,2^{6}\cdot 71^{4}$ |
$2$ |
$14.1710382773$ |
$34.292743267870684$ |
|
|
|
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$76.3102907518$ |
| 8.2.1673730583.1 |
$x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} + 2 x^{3} - 4 x^{2} + 3 x - 4$ |
$8$ |
[2,3] |
$-\,7^{3}\cdot 47^{4}$ |
$2$ |
$14.2220006441$ |
$34.45727470891496$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$89.5260039753$ |
| 8.2.1673730583.2 |
$x^{8} - 4 x^{7} + 12 x^{6} - 22 x^{5} + 32 x^{4} - 32 x^{3} + 8 x^{2} + 5 x - 2$ |
$8$ |
[2,3] |
$-\,7^{3}\cdot 47^{4}$ |
$2$ |
$14.2220006441$ |
$34.45727470891496$ |
|
|
? |
$\textrm{GL(2,3)}$ (as 8T23) |
trivial |
$2$ |
$4$ |
$89.5260039753$ |
