| 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 |
| 8.0.418195493.1 |
$x^{8} - x^{7} + 3 x^{6} - 3 x^{5} + 2 x^{4} - 2 x^{3} + 5 x^{2} + 5 x + 1$ |
$8$ |
[0,4] |
$53^{5}$ |
$1$ |
$11.9583818909$ |
$27.346173066268356$ |
|
|
✓ |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$6.97332028251$ |
| 8.2.1997331875.1 |
$x^{8} - x^{6} - 3 x^{5} - x^{4} + 4 x^{3} + 4 x^{2} - 2 x - 1$ |
$8$ |
[2,3] |
$-\,5^{4}\cdot 7^{4}\cdot 11^{3}$ |
$3$ |
$14.5397279001$ |
$35.48747003412059$ |
|
|
✓ |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$42.4915104951$ |
| 8.0.2449456192.1 |
$x^{8} - 2 x^{7} - 2 x^{6} + 4 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 7$ |
$8$ |
[0,4] |
$2^{6}\cdot 337^{3}$ |
$2$ |
$14.915358953711568$ |
$33.25375256866954$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$29.755767084265077$ |
| 8.0.3036027392.1 |
$x^{8} - x^{7} + 2 x^{6} - 4 x^{5} + 3 x^{4} + 5 x^{3} + 3 x^{2} - 9 x + 8$ |
$8$ |
[0,4] |
$2^{9}\cdot 181^{3}$ |
$2$ |
$15.3210387395$ |
$38.05259518088089$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$29.1660905095$ |
| 8.2.3157114563.1 |
$x^{8} - 2 x^{7} + x^{6} + x^{5} + 7 x^{4} - 14 x^{3} + 7 x^{2} + 7 x + 1$ |
$8$ |
[2,3] |
$-\,3^{6}\cdot 163^{3}$ |
$2$ |
$15.3961201875$ |
$32.7382060432246$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$4$ |
$26.3306012182$ |
| 8.0.3215578176.1 |
$x^{8} - 2 x^{7} - 2 x^{6} + 10 x^{5} + 4 x^{4} - 14 x^{3} - 2 x^{2} + 10 x + 4$ |
$8$ |
[0,4] |
$2^{6}\cdot 3^{6}\cdot 41^{3}$ |
$3$ |
$15.431473026749007$ |
$29.74258520496421$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$52.7514007724994$ |
| 8.2.4306878883.1 |
$x^{8} - x^{7} + 5 x^{6} - 3 x^{5} + 13 x^{4} - 8 x^{3} - 2 x^{2} + 5 x - 1$ |
$8$ |
[2,3] |
$-\,1627^{3}$ |
$1$ |
$16.005540046384944$ |
$40.33608806019741$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$4$ |
$29.641858304533457$ |
| 8.0.4913000000.1 |
$x^{8} - 3 x^{7} + 10 x^{6} - 16 x^{5} + 30 x^{4} - 32 x^{3} + 36 x^{2} - 20 x + 12$ |
$8$ |
[0,4] |
$2^{6}\cdot 5^{6}\cdot 17^{3}$ |
$3$ |
$16.271153625152323$ |
$29.67340047027482$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$74.2675947312699$ |
| 8.0.4991443829.1 |
$x^{8} - x^{7} + x^{6} + 4 x^{5} - x^{4} - 3 x^{3} + 5 x^{2} - 2 x + 1$ |
$8$ |
[0,4] |
$1709^{3}$ |
$1$ |
$16.3034033609$ |
$41.340053217188775$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$19.6849976235$ |
| 8.0.6759804861.1 |
$x^{8} - x^{7} + 4 x^{6} - 8 x^{5} + 14 x^{4} - 23 x^{3} + 26 x^{2} - 17 x + 5$ |
$8$ |
[0,4] |
$3^{8}\cdot 101^{3}$ |
$2$ |
$16.933306520560087$ |
$43.48328638861351$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$73.12433002647697$ |
| 8.0.7727161833.1 |
$x^{8} - 2 x^{6} - 2 x^{5} + 6 x^{4} + 8 x^{3} + 5 x^{2} + x + 1$ |
$8$ |
[0,4] |
$3^{3}\cdot 659^{3}$ |
$2$ |
$17.2187849082$ |
$44.46346815083142$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$44.7126640615$ |
| 8.2.9460870875.1 |
$x^{8} - 2 x^{7} - 2 x^{6} + 10 x^{5} + 7 x^{4} - 11 x^{3} - 11 x^{2} - 2 x + 1$ |
$8$ |
[2,3] |
$-\,3^{6}\cdot 5^{3}\cdot 47^{3}$ |
$3$ |
$17.660028064$ |
$39.30927252170212$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$4$ |
$56.5468183621$ |
| 8.2.10330523392.1 |
$x^{8} - 2 x^{7} + 7 x^{4} - 14 x^{2} + 8 x + 5$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 7^{9}$ |
$2$ |
$17.855223496$ |
$31.486213082006973$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$114.493755475$ |
| 8.2.10618817472.1 |
$x^{8} - 2 x^{7} - 2 x^{6} + 12 x^{5} - 20 x^{4} + 14 x^{3} - 6 x + 4$ |
$8$ |
[2,3] |
$-\,2^{6}\cdot 3^{4}\cdot 127^{3}$ |
$3$ |
$17.916761783569935$ |
$35.358041271479394$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$4$ |
$100.20272216474939$ |
| 8.0.10720659157.1 |
$x^{8} - x^{7} + 2 x^{6} - 5 x^{5} + 6 x^{4} + 2 x^{3} - 2 x^{2} - 5 x + 3$ |
$8$ |
[0,4] |
$13^{3}\cdot 47^{4}$ |
$2$ |
$17.938151437$ |
$46.95735016975237$ |
|
|
✓ |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$51.0317632328$ |
| 8.0.11806312448.1 |
$x^{8} - 4 x^{7} + 7 x^{6} - 14 x^{4} + 28 x^{2} - 28 x + 14$ |
$8$ |
[0,4] |
$2^{11}\cdot 7^{8}$ |
$2$ |
$18.155753765114135$ |
$39.646073744210554$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$102.57620515080315$ |
| 8.2.12057493063.2 |
$x^{8} - 3 x^{7} + 7 x^{4} - 7 x^{2} - x - 4$ |
$8$ |
[2,3] |
$-\,7^{7}\cdot 11^{4}$ |
$2$ |
$18.2035934486$ |
$43.64824030438486$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$174.359085829$ |
| 8.2.13875904827.1 |
$x^{8} - 2 x^{7} - x^{6} + 4 x^{5} - 2 x^{4} + 8 x^{3} - 2 x^{2} - 2 x - 1$ |
$8$ |
[2,3] |
$-\,3^{9}\cdot 89^{3}$ |
$2$ |
$18.5260428427$ |
$49.02040391510457$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$96.0312241004$ |
| 8.2.14270199808.1 |
$x^{8} + 4 x^{6} - 8 x^{5} + 6 x^{4} - 8 x^{3} + 2 x^{2} - 4 x - 1$ |
$8$ |
[2,3] |
$-\,2^{11}\cdot 191^{3}$ |
$2$ |
$18.5910430594$ |
$49.24986098871152$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$201.418901383$ |
| 8.0.14952627621.1 |
$x^{8} - 2 x^{7} - 3 x^{5} + 6 x^{4} + 3 x^{3} - 3 x^{2} + 3$ |
$8$ |
[0,4] |
$3^{6}\cdot 29^{5}$ |
$2$ |
$18.6999174655$ |
$42.42541250889309$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$138.710667345$ |
| 8.2.15748562368.1 |
$x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} - 2 x^{4} + 6 x^{3} - 4 x^{2} - 6 x + 8$ |
$8$ |
[2,3] |
$-\,2^{6}\cdot 7^{5}\cdot 11^{4}$ |
$3$ |
$18.8215382369$ |
$45.34567461518055$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$326.893144729$ |
| 8.0.16209796869.1 |
$x^{8} + 7 x^{6} + 21 x^{4} + 21 x^{2} - 18 x + 21$ |
$8$ |
[0,4] |
$3^{9}\cdot 7^{7}$ |
$2$ |
$18.8895754267$ |
$45.85501413123854$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$81.6363811054$ |
| 8.2.16387080192.5 |
$x^{8} - 4 x^{6} - 8 x^{5} + 32 x^{3} + 16 x^{2} - 1$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 3^{6}\cdot 7^{3}$ |
$3$ |
$18.9152766421$ |
$30.46083139496543$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$214.385154851$ |
| 8.2.18624991792.1 |
$x^{8} - 2 x^{7} + x^{6} - 3 x^{5} - 6 x^{4} - 7 x^{3} - 6 x^{2} - 8 x - 2$ |
$8$ |
[2,3] |
$-\,2^{4}\cdot 11^{4}\cdot 43^{3}$ |
$3$ |
$19.2203818383$ |
$51.48523290031809$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$331.577214856$ |
| 8.2.21562200000.1 |
$x^{8} - x^{7} - 4 x^{5} + 8 x^{3} + 2 x^{2} - 2$ |
$8$ |
[2,3] |
$-\,2^{6}\cdot 3^{4}\cdot 5^{5}\cdot 11^{3}$ |
$4$ |
$19.5754445621$ |
$39.78851074474477$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$440.62894485$ |
| 8.2.22105827328.1 |
$x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} + 4 x^{4} - 4 x^{3} - 8 x + 1$ |
$8$ |
[2,3] |
$-\,2^{11}\cdot 13^{3}\cdot 17^{3}$ |
$3$ |
$19.6364667561$ |
$52.97664636308887$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$265.603432969$ |
| 8.0.23328000000.1 |
$x^{8} - 3 x^{7} + 5 x^{6} - 20 x^{5} + 60 x^{4} - 100 x^{3} + 100 x^{2} - 60 x + 20$ |
$8$ |
[0,4] |
$2^{11}\cdot 3^{6}\cdot 5^{6}$ |
$3$ |
$19.7689994046$ |
$36.30500888940354$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$359.186925477$ |
| 8.0.24616775429.1 |
$x^{8} - x^{7} + x^{6} - 4 x^{4} + 4 x^{2} - x + 1$ |
$8$ |
[0,4] |
$2909^{3}$ |
$1$ |
$19.9023287388$ |
$53.9351462406472$ |
|
|
✓ |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$27.2352943712$ |
| 8.0.29019350017.1 |
$x^{8} - x^{7} - 4 x^{6} + 3 x^{5} + 5 x^{4} + 2 x^{3} + 4 x^{2} - 3 x + 1$ |
$8$ |
[0,4] |
$7^{3}\cdot 439^{3}$ |
$2$ |
$20.3158950247$ |
$55.4346462061408$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$101.918497287$ |
| 8.2.29076046875.1 |
$x^{8} - x^{7} - 6 x^{6} - 4 x^{5} + 30 x^{4} - x^{3} - 24 x^{2} - 9 x - 1$ |
$8$ |
[2,3] |
$-\,3^{3}\cdot 5^{6}\cdot 41^{3}$ |
$3$ |
$20.3208523378$ |
$44.06255762372035$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$4$ |
$88.3525081897$ |
| 8.0.29132817533.1 |
$x^{8} - 2 x^{7} + 4 x^{6} - 7 x^{4} + 3 x^{3} + 5 x^{2} - 4 x + 1$ |
$8$ |
[0,4] |
$17^{3}\cdot 181^{3}$ |
$2$ |
$20.3258076363$ |
$55.47071299343465$ |
|
|
✓ |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$22.3914517281$ |
| 8.2.31637227888.1 |
$x^{8} - x^{7} - 7 x^{6} + 7 x^{2} - 27 x - 1$ |
$8$ |
[2,3] |
$-\,2^{4}\cdot 7^{11}$ |
$2$ |
$20.5364230801$ |
$42.58971942746583$ |
|
|
|
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$229.140511319$ |
| 8.0.32891033664.1 |
$x^{8} - 2 x^{7} + 4 x^{6} + 16 x^{5} - 26 x^{4} - 32 x^{3} + 34 x^{2} + 10 x + 4$ |
$8$ |
[0,4] |
$2^{6}\cdot 3^{6}\cdot 89^{3}$ |
$3$ |
$20.63643577889669$ |
$43.820950092124605$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$193.93807618561115$ |
| 8.2.32941720000.1 |
$x^{8} - 3 x^{7} + 14 x^{4} - 14 x^{2} - 10 x + 2$ |
$8$ |
[2,3] |
$-\,2^{6}\cdot 5^{4}\cdot 7^{7}$ |
$3$ |
$20.6404082931$ |
$46.742362912802406$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$588.295128619$ |
| 8.2.37797742016.1 |
$x^{8} - x^{7} + 4 x^{5} - 4 x^{3} - 20 x^{2} - 16 x - 20$ |
$8$ |
[2,3] |
$-\,2^{6}\cdot 839^{3}$ |
$2$ |
$20.9982562875$ |
$52.4694716455353$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$4$ |
$193.130709722$ |
| 8.2.41135081408.1 |
$x^{8} - 2 x^{7} + 2 x^{4} + 2 x^{3} - 6 x^{2} + 2 x + 4$ |
$8$ |
[2,3] |
$-\,2^{6}\cdot 863^{3}$ |
$2$ |
$21.2215228313$ |
$53.21463754399128$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$4$ |
$216.518096952$ |
| 8.2.47756222027.1 |
$x^{8} - 2 x^{7} + 2 x^{6} + 3 x^{5} - 5 x^{4} - x^{3} + 7 x^{2} + x - 2$ |
$8$ |
[2,3] |
$-\,17^{4}\cdot 83^{3}$ |
$2$ |
$21.6211477874$ |
$60.23353156190304$ |
|
|
✓ |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$359.553162293$ |
| 8.0.52437868857.5 |
$x^{8} - 3 x^{7} + 9 x^{6} - 21 x^{5} + 42 x^{4} - 45 x^{3} + 57 x^{2} - 75 x + 36$ |
$8$ |
[0,4] |
$3^{6}\cdot 11^{4}\cdot 17^{3}$ |
$3$ |
$21.8753811681$ |
$35.06564879404867$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[6]$ |
$2$ |
$3$ |
$60.8321692416$ |
| 8.2.53971714048.5 |
$x^{8} - 4 x^{7} + 14 x^{4} - 8 x + 4$ |
$8$ |
[2,3] |
$-\,2^{16}\cdot 7^{7}$ |
$2$ |
$21.9543597473$ |
$39.62201944814752$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$229.834600006$ |
| 8.0.62587167125.1 |
$x^{8} - 2 x^{7} + 8 x^{6} - 13 x^{5} + 22 x^{4} - 27 x^{3} + 22 x^{2} + 7 x + 3$ |
$8$ |
[0,4] |
$5^{3}\cdot 7^{5}\cdot 31^{3}$ |
$3$ |
$22.3645756308$ |
$63.01068715206115$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$305.113854773$ |
| 8.2.63406466827.3 |
$x^{8} - x^{7} + 3 x^{6} - 10 x^{5} + 16 x^{4} - 33 x^{3} + 47 x^{2} - 33 x + 11$ |
$8$ |
[2,3] |
$-\,11^{4}\cdot 163^{3}$ |
$2$ |
$22.40096328434412$ |
$42.34383071948026$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$4$ |
$68.32993243131573$ |
| 8.0.65477433301.2 |
$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 49 x^{4} - 56 x^{3} + 63 x^{2} - 36 x + 32$ |
$8$ |
[0,4] |
$7^{7}\cdot 43^{3}$ |
$2$ |
$22.4911393236$ |
$57.86809387704$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$444.357525087$ |
| 8.0.67832083125.1 |
$x^{8} - x^{7} + 3 x^{6} - 8 x^{5} + 5 x^{4} + 9 x^{3} + 10 x^{2} - 19 x + 6$ |
$8$ |
[0,4] |
$3^{6}\cdot 5^{4}\cdot 53^{3}$ |
$3$ |
$22.590684709216468$ |
$41.743034080060816$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[6]$ |
$2$ |
$3$ |
$151.34946108760104$ |
| 8.2.72313663744.1 |
$x^{8} - x^{7} + 7 x^{6} + 4 x - 4$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 7^{10}$ |
$2$ |
$22.7720718638$ |
$48.8811899243406$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$607.517526659$ |
| 8.0.75622356992.1 |
$x^{8} - x^{7} + 7 x^{6} - 4 x + 4$ |
$8$ |
[0,4] |
$2^{11}\cdot 7^{5}\cdot 13^{3}$ |
$3$ |
$22.8997782127$ |
$65.0291935520838$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$378.488413926$ |
| 8.0.82644187136.1 |
$x^{8} - 3 x^{7} + 7 x^{6} - 14 x^{5} + 28 x^{4} - 14 x^{3} + 56 x^{2} + 26 x + 6$ |
$8$ |
[0,4] |
$2^{11}\cdot 7^{9}$ |
$2$ |
$23.1553600868$ |
$49.981247768872755$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$3$ |
$736.710228442$ |
| 8.0.83167327125.1 |
$x^{8} - x^{7} + 4 x^{6} - 4 x^{5} - 5 x^{4} - x^{3} + 7 x^{2} + 5 x + 1$ |
$8$ |
[0,4] |
$3^{6}\cdot 5^{3}\cdot 97^{3}$ |
$3$ |
$23.1736313052$ |
$56.47184082693539$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
$[3]$ |
$2$ |
$3$ |
$107.259780206$ |
| 8.2.86452249968.1 |
$x^{8} - 21 x^{4} + 28 x^{2} - 54 x - 63$ |
$8$ |
[2,3] |
$-\,2^{4}\cdot 3^{8}\cdot 7^{7}$ |
$3$ |
$23.2861149809$ |
$60.61125636630689$ |
|
|
|
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$652.547578294$ |
| 8.2.86452249968.2 |
$x^{8} - 3 x^{7} + 7 x^{6} - 7 x^{2} - 9 x - 1$ |
$8$ |
[2,3] |
$-\,2^{4}\cdot 3^{8}\cdot 7^{7}$ |
$3$ |
$23.2861149809$ |
$60.61125636630689$ |
|
|
|
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$712.991259667$ |
| 8.2.88941805312.1 |
$x^{8} - 2 x^{7} + 2 x^{6} + 6 x^{5} + 7 x^{4} + 6 x^{3} - 8 x^{2} + 2 x + 5$ |
$8$ |
[2,3] |
$-\,2^{8}\cdot 19^{3}\cdot 37^{3}$ |
$3$ |
$23.3688985146$ |
$59.522247876776156$ |
|
|
? |
$\PGL(2,7)$ (as 8T43) |
trivial |
$2$ |
$4$ |
$343.311229673$ |
