| 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 |
| 8.0.31443200.1 |
$x^{8} - 2 x^{7} + 3 x^{6} - 6 x^{5} + 13 x^{4} - 20 x^{3} + 21 x^{2} - 14 x + 5$ |
$8$ |
[0,4] |
$2^{8}\cdot 5^{2}\cdot 17^{3}$ |
$3$ |
$8.65348095632$ |
$18.439088914585774$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$5.01008972984$ |
| 8.0.31443200.2 |
$x^{8} - x^{6} - 4 x^{5} - 2 x^{4} + 4 x^{3} + 12 x^{2} + 6 x + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 5^{2}\cdot 17^{3}$ |
$3$ |
$8.65348095632$ |
$18.439088914585774$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$5.01008972984$ |
| 8.2.40960000.1 |
$x^{8} - 4 x^{6} + 5 x^{4} - 2 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 5^{4}$ |
$2$ |
$8.94427191$ |
$13.793952964992862$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$3.82805440637$ |
| 8.2.40960000.2 |
$x^{8} - x^{4} - 1$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 5^{4}$ |
$2$ |
$8.94427191$ |
$13.793952964992862$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$3.82805440637$ |
| 8.0.70304000.1 |
$x^{8} + x^{6} - 2 x^{5} + 5 x^{4} - 2 x^{3} + x^{2} + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 5^{3}\cdot 13^{3}$ |
$3$ |
$9.56913109595$ |
$16.1245154965971$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$9.3924676172$ |
| 8.0.70304000.2 |
$x^{8} - 4 x^{7} + 7 x^{6} - 2 x^{5} - 8 x^{4} + 8 x^{3} + 2 x^{2} - 4 x + 1$ |
$8$ |
[0,4] |
$2^{8}\cdot 5^{3}\cdot 13^{3}$ |
$3$ |
$9.56913109595$ |
$16.1245154965971$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$9.3924676172$ |
| 8.2.131274675.1 |
$x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + x^{4} - 5 x^{3} + 4 x^{2} - 2 x + 1$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 5^{2}\cdot 7^{4}$ |
$3$ |
$10.3460034518$ |
$15.470883652211068$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$9.89882481645$ |
| 8.2.131274675.2 |
$x^{8} - x^{7} + x^{6} - 4 x^{5} + x^{4} - 4 x^{3} + x^{2} - x + 1$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 5^{2}\cdot 7^{4}$ |
$3$ |
$10.3460034518$ |
$15.470883652211068$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$9.89882481645$ |
| 8.0.143327232.1 |
$x^{8} - 3 x^{4} + 3$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{7}$ |
$2$ |
$10.4602265145$ |
$16.131874566820663$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$6$ |
$3$ |
$24.0787745359$ |
| 8.0.143327232.2 |
$x^{8} - 6 x^{6} + 15 x^{4} - 12 x^{2} + 3$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{7}$ |
$2$ |
$10.4602265145$ |
$16.131874566820663$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$6$ |
$3$ |
$24.0787745359$ |
| 8.0.212556032.1 |
$x^{8} - 2 x^{7} + 5 x^{6} - 10 x^{5} + 23 x^{4} - 36 x^{3} + 51 x^{2} - 48 x + 17$ |
$8$ |
[0,4] |
$2^{8}\cdot 13^{2}\cdot 17^{3}$ |
$3$ |
$10.9883928908$ |
$29.732137494637012$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$14.7109421373$ |
| 8.0.212556032.2 |
$x^{8} - 4 x^{7} + x^{6} + 8 x^{5} + 5 x^{4} - 14 x^{3} - 13 x^{2} + 4 x + 17$ |
$8$ |
[0,4] |
$2^{8}\cdot 13^{2}\cdot 17^{3}$ |
$3$ |
$10.9883928908$ |
$29.732137494637012$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$14.7109421373$ |
| 8.2.342102016.1 |
$x^{8} - 4 x^{6} + 5 x^{4} - 2 x^{2} - 4$ |
$8$ |
[2,3] |
$-\,2^{12}\cdot 17^{4}$ |
$2$ |
$11.6619037897$ |
$19.61290618356907$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$21.2794975552$ |
| 8.2.342102016.2 |
$x^{8} - 8 x^{4} - 1$ |
$8$ |
[2,3] |
$-\,2^{12}\cdot 17^{4}$ |
$2$ |
$11.6619037897$ |
$19.61290618356907$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$21.2794975552$ |
| 8.2.359661568.1 |
$x^{8} - 4 x^{5} + 8 x^{3} + 4 x^{2} + 4 x + 1$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 7^{3}$ |
$2$ |
$11.7350989029$ |
$14.966629547095765$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$21.8589719581$ |
| 8.2.359661568.2 |
$x^{8} - 4 x^{6} - 4 x^{5} + 2 x^{4} - 2$ |
$8$ |
[2,3] |
$-\,2^{20}\cdot 7^{3}$ |
$2$ |
$11.7350989029$ |
$14.966629547095765$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$21.8589719581$ |
| 8.2.518711875.1 |
$x^{8} - 6 x^{6} - x^{5} + 4 x^{4} + 13 x^{3} + 9 x^{2} + 10 x - 5$ |
$8$ |
[2,3] |
$-\,5^{4}\cdot 11^{2}\cdot 19^{3}$ |
$3$ |
$12.2847344703$ |
$32.32645975048923$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$20.8871501936$ |
| 8.2.518711875.2 |
$x^{8} - x^{7} - 6 x^{6} + 8 x^{5} + 11 x^{4} - 15 x^{3} - 29 x^{2} + 55 x - 25$ |
$8$ |
[2,3] |
$-\,5^{4}\cdot 11^{2}\cdot 19^{3}$ |
$3$ |
$12.2847344703$ |
$32.32645975048923$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
$[2]$ |
$2$ |
$4$ |
$10.4435750968$ |
| 8.0.693388917.1 |
$x^{8} - 3 x^{7} - 2 x^{6} + 15 x^{5} - 20 x^{4} + 15 x^{3} + 30 x^{2} - 108 x + 81$ |
$8$ |
[0,4] |
$3^{4}\cdot 13^{2}\cdot 37^{3}$ |
$3$ |
$12.7386120628$ |
$37.986839826445156$ |
|
|
|
$Z_8 : Z_8^\times$ (as 8T15) |
$[2]$ |
$6$ |
$3$ |
$52.2538386151$ |
| 8.0.693388917.2 |
$x^{8} - 3 x^{7} - 5 x^{6} + 27 x^{5} - 11 x^{4} - 60 x^{3} + 69 x^{2} - 18 x + 9$ |
$8$ |
[0,4] |
$3^{4}\cdot 13^{2}\cdot 37^{3}$ |
$3$ |
$12.7386120628$ |
$37.986839826445156$ |
|
|
|
$Z_8 : Z_8^\times$ (as 8T15) |
$[2]$ |
$6$ |
$3$ |
$52.2538386151$ |
| 8.2.1024000000.1 |
$x^{8} - 15 x^{4} - 50 x^{2} - 25$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 5^{6}$ |
$2$ |
$13.3748060995$ |
$20.626770754424918$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$21.4714315024$ |
| 8.2.1024000000.2 |
$x^{8} - 5 x^{4} - 25$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 5^{6}$ |
$2$ |
$13.3748060995$ |
$20.626770754424918$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$21.4714315024$ |
| 8.0.1057749248.1 |
$x^{8} - 2 x^{7} - 3 x^{6} + 10 x^{5} - 6 x^{4} - 28 x^{3} + 46 x^{2} + 110 x + 53$ |
$8$ |
[0,4] |
$2^{8}\cdot 17^{3}\cdot 29^{2}$ |
$3$ |
$13.4291288449$ |
$44.40720662234904$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$32.8153952379$ |
| 8.0.1057749248.2 |
$x^{8} - 2 x^{7} - 7 x^{6} + 14 x^{5} + 15 x^{4} - 48 x^{3} - 17 x^{2} + 96 x + 73$ |
$8$ |
[0,4] |
$2^{8}\cdot 17^{3}\cdot 29^{2}$ |
$3$ |
$13.4291288449$ |
$44.40720662234904$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$32.8153952379$ |
| 8.2.1097440000.1 |
$x^{8} + 5 x^{6} + 3 x^{4} - 15 x^{2} - 19$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 5^{4}\cdot 19^{3}$ |
$3$ |
$13.4911073552$ |
$27.568097504180443$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
$[2]$ |
$2$ |
$4$ |
$16.0824832914$ |
| 8.2.1097440000.2 |
$x^{8} - 5 x^{6} + 3 x^{4} + 15 x^{2} - 19$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 5^{4}\cdot 19^{3}$ |
$3$ |
$13.4911073552$ |
$27.568097504180443$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$32.1649665828$ |
| 8.2.1119744000.1 |
$x^{8} - 2 x^{7} + x^{6} + 4 x^{5} - 5 x^{4} + 4 x^{3} + x^{2} - 2 x + 1$ |
$8$ |
[2,3] |
$-\,2^{12}\cdot 3^{7}\cdot 5^{3}$ |
$3$ |
$13.5250800122$ |
$16.539070313490214$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$38.0733264654$ |
| 8.2.1119744000.2 |
$x^{8} - 2 x^{7} + x^{6} - 8 x^{5} + x^{4} - 8 x^{3} + x^{2} - 2 x + 1$ |
$8$ |
[2,3] |
$-\,2^{12}\cdot 3^{7}\cdot 5^{3}$ |
$3$ |
$13.5250800122$ |
$16.539070313490214$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$38.0733264654$ |
| 8.0.1287913472.1 |
$x^{8} - 10 x^{6} + 42 x^{4} - 84 x^{2} + 68$ |
$8$ |
[0,4] |
$2^{18}\cdot 17^{3}$ |
$2$ |
$13.763720213$ |
$27.736837922354518$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$47.1972100966$ |
| 8.0.1287913472.2 |
$x^{8} - 2 x^{6} + 10 x^{4} - 4 x^{2} + 68$ |
$8$ |
[0,4] |
$2^{18}\cdot 17^{3}$ |
$2$ |
$13.763720213$ |
$27.736837922354518$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$47.1972100966$ |
| 8.2.1344252672.1 |
$x^{8} + 3 x^{6} - 6 x^{2} - 3$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 3^{7}\cdot 7^{4}$ |
$3$ |
$13.8375790073$ |
$19.5692919025805$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$36.1393783434$ |
| 8.2.1344252672.2 |
$x^{8} - 3 x^{6} + 6 x^{2} - 3$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 3^{7}\cdot 7^{4}$ |
$3$ |
$13.8375790073$ |
$19.5692919025805$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$36.1393783434$ |
| 8.2.1517535243.1 |
$x^{8} - x^{7} + 4 x^{6} - x^{5} + 7 x^{4} - 10 x^{3} - 8 x^{2} + 14 x - 5$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 7^{4}\cdot 17^{2}$ |
$3$ |
$14.0489008742$ |
$28.52689992503314$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$37.4830482781$ |
| 8.2.1517535243.2 |
$x^{8} - x^{7} - 5 x^{6} + 8 x^{5} + 4 x^{4} - 25 x^{3} - 5 x^{2} + 11 x - 5$ |
$8$ |
[2,3] |
$-\,3^{7}\cdot 7^{4}\cdot 17^{2}$ |
$3$ |
$14.0489008742$ |
$28.52689992503314$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$37.4830482781$ |
| 8.0.1686616064.1 |
$x^{8} + 7 x^{6} + 21 x^{4} + 21 x^{2} + 14$ |
$8$ |
[0,4] |
$2^{11}\cdot 7^{7}$ |
$2$ |
$14.2356410587$ |
$21.954359747298287$ |
|
|
|
$Z_8 : Z_8^\times$ (as 8T15) |
$[2]$ |
$2$ |
$3$ |
$84.6660090557$ |
| 8.0.1686616064.2 |
$x^{8} - 2 x^{7} + 7 x^{4} - 14 x^{3} + 12 x + 4$ |
$8$ |
[0,4] |
$2^{11}\cdot 7^{7}$ |
$2$ |
$14.2356410587$ |
$21.954359747298287$ |
|
|
|
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$3$ |
$169.332018111$ |
| 8.0.1715072373.1 |
$x^{8} - 2 x^{7} - 2 x^{5} + 7 x^{4} - 9 x^{3} + 27 x^{2} - 42 x + 21$ |
$8$ |
[0,4] |
$3^{6}\cdot 7^{3}\cdot 19^{3}$ |
$3$ |
$14.2654444318$ |
$26.28855781932478$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
$[2]$ |
$6$ |
$3$ |
$66.9051081171$ |
| 8.0.1715072373.2 |
$x^{8} - 4 x^{7} + x^{6} + 11 x^{5} - 14 x^{4} + 5 x^{3} + 28 x^{2} - 28 x + 49$ |
$8$ |
[0,4] |
$3^{6}\cdot 7^{3}\cdot 19^{3}$ |
$3$ |
$14.2654444318$ |
$26.28855781932478$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
$[2]$ |
$6$ |
$3$ |
$66.9051081171$ |
| 8.0.1721829632.1 |
$x^{8} - 2 x^{7} - 5 x^{6} + 24 x^{5} + 16 x^{4} - 72 x^{3} - 36 x^{2} + 74 x + 53$ |
$8$ |
[0,4] |
$2^{8}\cdot 17^{3}\cdot 37^{2}$ |
$3$ |
$14.2724579516$ |
$50.15974481593781$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$42.3287213263$ |
| 8.0.1721829632.2 |
$x^{8} - 4 x^{7} + 3 x^{6} + 9 x^{4} + 28 x^{3} - 89 x^{2} - 36 x + 149$ |
$8$ |
[0,4] |
$2^{8}\cdot 17^{3}\cdot 37^{2}$ |
$3$ |
$14.2724579516$ |
$50.15974481593781$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$4$ |
$3$ |
$42.3287213263$ |
| 8.0.1809375813.1 |
$x^{8} - 3 x^{7} + 11 x^{6} - 33 x^{5} + 57 x^{4} - 108 x^{3} + 149 x^{2} - 90 x + 127$ |
$8$ |
[0,4] |
$3^{6}\cdot 7^{2}\cdot 37^{3}$ |
$3$ |
$14.3612121412$ |
$36.68519425437213$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
$[2]$ |
$6$ |
$3$ |
$43.5848861293$ |
| 8.0.1809375813.2 |
$x^{8} - 3 x^{7} + 14 x^{6} - 27 x^{5} + 66 x^{4} - 57 x^{3} + 116 x^{2} + 6 x + 13$ |
$8$ |
[0,4] |
$3^{6}\cdot 7^{2}\cdot 37^{3}$ |
$3$ |
$14.3612121412$ |
$36.68519425437213$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
$[2]$ |
$6$ |
$3$ |
$43.5848861293$ |
| 8.0.1820786688.1 |
$x^{8} + 2 x^{6} - x^{4} - 8 x^{2} + 7$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 7^{3}$ |
$3$ |
$14.3725021966$ |
$28.269191380048856$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$6$ |
$3$ |
$98.0889638577$ |
| 8.0.1820786688.2 |
$x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 7$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 7^{3}$ |
$3$ |
$14.3725021966$ |
$28.269191380048856$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$6$ |
$3$ |
$98.0889638577$ |
| 8.0.1820786688.3 |
$x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{2} + 7$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 7^{3}$ |
$3$ |
$14.3725021966$ |
$28.269191380048856$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$6$ |
$3$ |
$59.6275340257$ |
| 8.0.1820786688.4 |
$x^{8} - 2 x^{6} - x^{4} + 8 x^{2} + 7$ |
$8$ |
[0,4] |
$2^{16}\cdot 3^{4}\cdot 7^{3}$ |
$3$ |
$14.3725021966$ |
$28.269191380048856$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$6$ |
$3$ |
$59.6275340257$ |
| 8.2.1871773696.1 |
$x^{8} - 2 x^{6} - 8 x^{4} + 22 x^{2} - 9$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 13^{4}$ |
$2$ |
$14.4222051019$ |
$22.24208083433589$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$61.0598314827$ |
| 8.2.1871773696.2 |
$x^{8} - 3 x^{4} - 1$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 13^{4}$ |
$2$ |
$14.4222051019$ |
$22.24208083433589$ |
|
|
✓ |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$61.0598314827$ |
| 8.2.1897443072.1 |
$x^{8} - 2 x^{7} + 7 x^{2} - 2 x - 3$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 3^{2}\cdot 7^{7}$ |
$3$ |
$14.4467811672$ |
$19.013033264982823$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$53.6212538308$ |
| 8.2.1897443072.2 |
$x^{8} - 2 x^{7} + 7 x^{6} - 14 x^{5} + 14 x^{4} - 14 x^{2} + 12 x - 3$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 3^{2}\cdot 7^{7}$ |
$3$ |
$14.4467811672$ |
$19.013033264982823$ |
|
|
? |
$Z_8 : Z_8^\times$ (as 8T15) |
trivial |
$2$ |
$4$ |
$53.6212538308$ |
