| 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 |
| 4.2.275.1 |
$x^{4} - x^{3} + 2 x - 1$ |
$4$ |
[2,1] |
$-\,5^{2}\cdot 11$ |
$2$ |
$4.07223819929$ |
$7.416198487095663$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$0.369184373575$ |
| 4.2.283.1 |
$x^{4} - x - 1$ |
$4$ |
[2,1] |
$-\,283$ |
$1$ |
$4.10153676581$ |
$16.822603841260722$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$0.37819933246$ |
| 4.2.331.1 |
$x^{4} - x^{3} + x^{2} + x - 1$ |
$4$ |
[2,1] |
$-\,331$ |
$1$ |
$4.26537283232$ |
$18.193405398660254$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$0.432203091185$ |
| 4.2.400.1 |
$x^{4} - x^{2} - 1$ |
$4$ |
[2,1] |
$-\,2^{4}\cdot 5^{2}$ |
$2$ |
$4.472135955$ |
$6.324555320336759$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$0.51069810943$ |
| 4.2.448.1 |
$x^{4} - 2 x^{3} + x^{2} - 2 x + 1$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 7$ |
$2$ |
$4.60065326758$ |
$7.483314773547883$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$0.557886846205$ |
| 4.2.475.1 |
$x^{4} - 2 x^{3} + 2 x^{2} - x - 1$ |
$4$ |
[2,1] |
$-\,5^{2}\cdot 19$ |
$2$ |
$4.66845742379$ |
$9.746794344808963$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$0.57693147741$ |
| 4.2.491.1 |
$x^{4} - x^{3} - x^{2} + 3 x - 1$ |
$4$ |
[2,1] |
$-\,491$ |
$1$ |
$4.70728369723$ |
$22.15851980616034$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$0.63355773488$ |
| 4.2.507.1 |
$x^{4} - x^{3} - x^{2} - x + 1$ |
$4$ |
[2,1] |
$-\,3\cdot 13^{2}$ |
$2$ |
$4.74517233601$ |
$6.244997998398398$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$0.649395711926$ |
| 4.2.563.1 |
$x^{4} - x^{3} + x^{2} - x - 1$ |
$4$ |
[2,1] |
$-\,563$ |
$1$ |
$4.87110059796$ |
$23.727621035409346$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$0.704267195983$ |
| 4.2.643.1 |
$x^{4} - x^{3} - 2 x + 1$ |
$4$ |
[2,1] |
$-\,643$ |
$1$ |
$5.03561760524$ |
$25.357444666211933$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$0.720184467868$ |
| 4.2.688.1 |
$x^{4} - 2 x - 1$ |
$4$ |
[2,1] |
$-\,2^{4}\cdot 43$ |
$2$ |
$5.12149920406$ |
$14.720951755301632$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$0.998743211853$ |
| 4.2.731.1 |
$x^{4} - x^{3} + 2 x^{2} - 1$ |
$4$ |
[2,1] |
$-\,17\cdot 43$ |
$2$ |
$5.19971265256$ |
$27.03701166919155$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$0.866830073687$ |
| 4.2.751.1 |
$x^{4} - 2 x^{3} + x^{2} - x - 1$ |
$4$ |
[2,1] |
$-\,751$ |
$1$ |
$5.23491921734$ |
$27.40437921208944$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.06794059626$ |
| 4.2.775.1 |
$x^{4} - x^{3} - 3 x - 1$ |
$4$ |
[2,1] |
$-\,5^{2}\cdot 31$ |
$2$ |
$5.27625073458$ |
$12.449899597988733$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$0.867905309463$ |
| 4.2.848.1 |
$x^{4} - x^{2} - 2 x + 1$ |
$4$ |
[2,1] |
$-\,2^{4}\cdot 53$ |
$2$ |
$5.39633575282$ |
$20.591260281974$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$0.992994938315$ |
| 4.2.976.1 |
$x^{4} - 2 x^{3} + 3 x^{2} - 1$ |
$4$ |
[2,1] |
$-\,2^{4}\cdot 61$ |
$2$ |
$5.58936478534$ |
$22.090722034374522$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$0.990699215006$ |
| 4.2.1024.1 |
$x^{4} - 2 x^{2} - 1$ |
$4$ |
[2,1] |
$-\,2^{10}$ |
$1$ |
$5.65685424949$ |
$6.727171322029716$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$1.34724203432$ |
| 4.2.1099.1 |
$x^{4} - x^{3} + x^{2} - 3 x + 1$ |
$4$ |
[2,1] |
$-\,7\cdot 157$ |
$2$ |
$5.75770517728$ |
$33.15116890850155$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.01244916602$ |
| 4.2.1107.1 |
$x^{4} - x^{3} - 2 x - 1$ |
$4$ |
[2,1] |
$-\,3^{3}\cdot 41$ |
$2$ |
$5.76815477594$ |
$23.06924557594881$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.28804807411$ |
| 4.2.1156.1 |
$x^{4} - x^{3} - 2 x^{2} - x + 1$ |
$4$ |
[2,1] |
$-\,2^{2}\cdot 17^{2}$ |
$2$ |
$5.83095189485$ |
$8.246211251235321$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$1.53512616922$ |
| 4.2.1192.1 |
$x^{4} - x^{3} + 2 x^{2} + x - 1$ |
$4$ |
[2,1] |
$-\,2^{3}\cdot 149$ |
$2$ |
$5.87582785684$ |
$34.52535300326414$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.5689723036$ |
| 4.2.1255.1 |
$x^{4} - x^{2} - 3 x - 1$ |
$4$ |
[2,1] |
$-\,5\cdot 251$ |
$2$ |
$5.95197271229$ |
$35.4259791678367$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.58232994952$ |
| 4.2.1323.1 |
$x^{4} - x^{3} - 3 x^{2} - x + 1$ |
$4$ |
[2,1] |
$-\,3^{3}\cdot 7^{2}$ |
$2$ |
$6.03100878452$ |
$6.03100878451909$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$1.35144993247$ |
| 4.2.1328.1 |
$x^{4} - 3 x^{2} - 2 x + 1$ |
$4$ |
[2,1] |
$-\,2^{4}\cdot 83$ |
$2$ |
$6.03669895858$ |
$18.2208671582886$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.3668896797$ |
| 4.2.1371.1 |
$x^{4} + 2 x^{2} - x - 1$ |
$4$ |
[2,1] |
$-\,3\cdot 457$ |
$2$ |
$6.08498292217$ |
$37.027017163147235$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.34876542307$ |
| 4.2.1375.1 |
$x^{4} - x^{3} + x^{2} + 4 x - 4$ |
$4$ |
[2,1] |
$-\,5^{3}\cdot 11$ |
$2$ |
$6.08941642815$ |
$11.08980336897316$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$1.33061932583$ |
| 4.2.1399.1 |
$x^{4} - x^{3} + x - 2$ |
$4$ |
[2,1] |
$-\,1399$ |
$1$ |
$6.11581625121$ |
$37.403208418530085$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.70613352231$ |
| 4.2.1423.1 |
$x^{4} - x^{3} + x^{2} - 2 x - 1$ |
$4$ |
[2,1] |
$-\,1423$ |
$1$ |
$6.14187855853$ |
$37.72267222772003$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.69769459199$ |
| 4.2.1424.1 |
$x^{4} + x^{2} - 2 x - 1$ |
$4$ |
[2,1] |
$-\,2^{4}\cdot 89$ |
$2$ |
$6.14295731128$ |
$26.68332812825267$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.47325330706$ |
| 4.2.1456.1 |
$x^{4} - 2 x^{2} - 2 x + 1$ |
$4$ |
[2,1] |
$-\,2^{4}\cdot 7\cdot 13$ |
$3$ |
$6.17718123878$ |
$21.415210999701447$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.76052319498$ |
| 4.2.1472.1 |
$x^{4} - 2 x^{3} + 2 x^{2} - 2$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 23$ |
$2$ |
$6.19408202936$ |
$13.564659966250536$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.95632588969$ |
| 4.2.1472.2 |
$x^{4} - 2 x^{3} + 3 x^{2} - 2 x - 1$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 23$ |
$2$ |
$6.19408202936$ |
$13.564659966250536$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$1.25951480215$ |
| 4.2.1475.1 |
$x^{4} - x^{3} - x^{2} + 5 x - 5$ |
$4$ |
[2,1] |
$-\,5^{2}\cdot 59$ |
$2$ |
$6.19723557236$ |
$17.175564037317667$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$1.23477697451$ |
| 4.2.1588.1 |
$x^{4} - x^{3} - 3 x^{2} + 2$ |
$4$ |
[2,1] |
$-\,2^{2}\cdot 397$ |
$2$ |
$6.31266328029$ |
$39.84971769034255$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.93358038312$ |
| 4.2.1600.1 |
$x^{4} - 2 x^{2} - 4$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 5^{2}$ |
$2$ |
$6.32455532034$ |
$8.94427190999916$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$1.41413886912$ |
| 4.2.1728.1 |
$x^{4} - 2 x^{3} - 2 x + 1$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 3^{3}$ |
$2$ |
$6.44741959094$ |
$6.4474195909412515$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$2.18995070591$ |
| 4.2.1732.1 |
$x^{4} - x^{3} + 3 x - 1$ |
$4$ |
[2,1] |
$-\,2^{2}\cdot 433$ |
$2$ |
$6.45114750206$ |
$41.617304093369626$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$2.05155660923$ |
| 4.2.1775.1 |
$x^{4} - 2 x^{3} + 3 x^{2} - 2 x - 4$ |
$4$ |
[2,1] |
$-\,5^{2}\cdot 71$ |
$2$ |
$6.49082035384$ |
$18.841443681416774$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$1.51169722096$ |
| 4.2.1791.1 |
$x^{4} - x^{3} - x^{2} - 2 x + 1$ |
$4$ |
[2,1] |
$-\,3^{2}\cdot 199$ |
$2$ |
$6.50539836897$ |
$24.43358344574123$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$2.01913849925$ |
| 4.2.1792.1 |
$x^{4} - 2 x^{2} - 4 x - 2$ |
$4$ |
[2,1] |
$-\,2^{8}\cdot 7$ |
$2$ |
$6.50630624679$ |
$10.583005244258363$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$1.98905564548$ |
| 4.2.1823.1 |
$x^{4} - x^{3} + 3 x - 2$ |
$4$ |
[2,1] |
$-\,1823$ |
$1$ |
$6.53426385145$ |
$42.69660408041839$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$2.28192620748$ |
| 4.2.1856.1 |
$x^{4} - 2 x^{3} + x^{2} - 2 x - 1$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 29$ |
$2$ |
$6.56363606982$ |
$21.540659228538015$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.76145911782$ |
| 4.2.1879.1 |
$x^{4} - x^{3} - 2 x^{2} - 3 x + 1$ |
$4$ |
[2,1] |
$-\,1879$ |
$1$ |
$6.58387679133$ |
$43.347433603386484$ |
|
|
|
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.6589571085$ |
| 4.2.1927.1 |
$x^{4} - 2 x^{3} + x^{2} - 3 x + 1$ |
$4$ |
[2,1] |
$-\,41\cdot 47$ |
$2$ |
$6.62552700812$ |
$43.89760813529594$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$2.07547726245$ |
| 4.2.1931.1 |
$x^{4} - 3 x + 1$ |
$4$ |
[2,1] |
$-\,1931$ |
$1$ |
$6.62896259496$ |
$43.94314508543966$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.39084475468$ |
| 4.2.1963.1 |
$x^{4} - x^{3} + 2 x^{2} - 2 x - 1$ |
$4$ |
[2,1] |
$-\,13\cdot 151$ |
$2$ |
$6.65625689361$ |
$44.30575583375144$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$1.47885191076$ |
| 4.2.1968.1 |
$x^{4} - 4 x^{2} - 2 x + 2$ |
$4$ |
[2,1] |
$-\,2^{4}\cdot 3\cdot 41$ |
$3$ |
$6.66049142524$ |
$24.897412647668382$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$2.39920685452$ |
| 4.2.1975.1 |
$x^{4} - 2 x^{3} + x^{2} - 5$ |
$4$ |
[2,1] |
$-\,5^{2}\cdot 79$ |
$2$ |
$6.66640623474$ |
$19.87460691435179$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$1.77633002997$ |
| 4.2.1984.1 |
$x^{4} - 2 x^{3} + x^{2} - 2$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 31$ |
$2$ |
$6.67398793096$ |
$15.748015748023622$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$3.00091979939$ |
| 4.2.1984.2 |
$x^{4} - 2 x^{3} + 2 x^{2} + 2 x - 1$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 31$ |
$2$ |
$6.67398793096$ |
$15.748015748023622$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$2$ |
$2.16801442334$ |
