| Label |
Polynomial |
Degree |
Signature |
Discriminant |
Ram. prime count |
Root discriminant |
Galois root discriminant |
CM field |
Galois |
Monogenic |
Galois group |
Class group |
Narrow class group |
Unit group torsion |
Unit group rank |
Regulator |
Unit signature rank |
Max $p$ |
| 4.2.38912.2 |
$x^{4} - 4 x^{2} - 16 x - 6$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 19$ |
$2$ |
$14.0449723423$ |
$29.323059968612448$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$8.6711009944$ |
$1$ |
$19$ |
| 4.0.92160.4 |
$x^{4} - 8 x^{2} + 40$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 5$ |
$3$ |
$17.4235086988$ |
$26.054222497305222$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$6.45240565626$ |
$0$ |
$5$ |
| 4.2.153600.1 |
$x^{4} - 4 x^{2} - 6$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 5^{2}$ |
$3$ |
$19.7969280154$ |
$26.054222497305222$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$23.4667084383$ |
$1$ |
$5$ |
| 4.4.256000.2 |
$x^{4} - 20 x^{2} + 90$ |
$4$ |
(4, 0) |
$2^{11}\cdot 5^{3}$ |
$2$ |
$22.4936530076$ |
$22.493653007613965$ |
|
✓ |
|
$C_4$ (as 4T1) |
$[2]$ |
$[4]$ |
$2$ |
$3$ |
$23.1560954216$ |
$3$ |
$5$ |
| 4.2.301056.1 |
$x^{4} + 8 x^{2} - 8 x - 10$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 7^{2}$ |
$3$ |
$23.4240411198$ |
$42.63750749706666$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$27.256274086$ |
$1$ |
$7$ |
| 4.0.354304.4 |
$x^{4} + 8 x^{2} - 32 x + 40$ |
$4$ |
(0, 2) |
$2^{11}\cdot 173$ |
$2$ |
$24.3974262497$ |
$88.48212407767714$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$6.02614233716$ |
$0$ |
$173$ |
| 4.0.387072.4 |
$x^{4} + 24 x^{2} + 168$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{3}\cdot 7$ |
$3$ |
$24.942957063$ |
$40.571629338126115$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$11.1561076502$ |
$0$ |
$7$ |
| 4.0.534528.5 |
$x^{4} - 4 x^{2} + 58$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 29$ |
$3$ |
$27.0391238424$ |
$62.74687687563977$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$8.93772901978$ |
$0$ |
$29$ |
| 4.0.665600.3 |
$x^{4} + 12 x^{2} + 26$ |
$4$ |
(0, 2) |
$2^{11}\cdot 5^{2}\cdot 13$ |
$3$ |
$28.562967671$ |
$54.23618911516588$ |
✓ |
|
✓ |
$D_{4}$ (as 4T3) |
$[20]$ |
$[20]$ |
$2$ |
$1$ |
$3.63689291846$ |
$0$ |
$13$ |
| 4.0.976896.3 |
$x^{4} - 40 x^{2} + 424$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 53$ |
$3$ |
$31.4385192643$ |
$84.82640276089931$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$24.2779163769$ |
$0$ |
$53$ |
| 4.2.989184.1 |
$x^{4} + 12 x^{2} - 16 x - 6$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 7\cdot 23$ |
$4$ |
$31.5369195887$ |
$147.84479977340615$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$86.7806059419$ |
$1$ |
$23$ |
| 4.2.1398784.1 |
$x^{4} - 8 x^{2} - 8 x + 22$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 683$ |
$2$ |
$34.3904345888$ |
$175.80970285859186$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$45.7349091611$ |
$1$ |
$683$ |
| 4.0.1419264.1 |
$x^{4} - 20 x^{2} + 154$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 7\cdot 11$ |
$4$ |
$34.5156293585$ |
$102.24415217048806$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$40.5648295897$ |
$0$ |
$11$ |
| 4.0.1419264.2 |
$x^{4} - 4 x^{2} + 154$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 7\cdot 11$ |
$4$ |
$34.5156293585$ |
$102.24415217048806$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$12.7323796574$ |
$0$ |
$11$ |
| 4.0.1583104.2 |
$x^{4} + 8 x^{2} - 8 x + 22$ |
$4$ |
(0, 2) |
$2^{11}\cdot 773$ |
$2$ |
$35.4713209239$ |
$187.03472051696946$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$15.9752594874$ |
$0$ |
$773$ |
| 4.0.1714176.2 |
$x^{4} + 12 x^{2} + 186$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{3}\cdot 31$ |
$3$ |
$36.1837731989$ |
$85.3796314967612$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$26.9507481795$ |
$0$ |
$31$ |
| 4.0.1730560.2 |
$x^{4} + 24 x^{2} + 40$ |
$4$ |
(0, 2) |
$2^{11}\cdot 5\cdot 13^{2}$ |
$3$ |
$36.2699256496$ |
$54.23618911516588$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
$[20]$ |
$[20]$ |
$2$ |
$1$ |
$4.62487668255$ |
$0$ |
$13$ |
| 4.2.1775616.2 |
$x^{4} - 24 x^{2} - 72 x + 54$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 17^{2}$ |
$3$ |
$36.5037315911$ |
$77.03576441267185$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$47.6549191467$ |
$1$ |
$17$ |
| 4.2.1824768.1 |
$x^{4} + 12 x^{2} - 80 x - 102$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3^{4}\cdot 11$ |
$3$ |
$36.7537717633$ |
$96.5362676038149$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$74.5292074544$ |
$1$ |
$11$ |
| 4.0.1861632.5 |
$x^{4} + 56 x^{2} + 808$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 101$ |
$3$ |
$36.9380069676$ |
$117.09916609766924$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$12.1019243413$ |
$0$ |
$101$ |
| 4.4.1894400.2 |
$x^{4} - 52 x^{2} + 666$ |
$4$ |
(4, 0) |
$2^{11}\cdot 5^{2}\cdot 37$ |
$3$ |
$37.0994884429$ |
$91.49942234379802$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[4]$ |
$2$ |
$3$ |
$124.202990086$ |
$3$ |
$37$ |
| 4.0.1910784.2 |
$x^{4} - 8 x^{2} - 72 x + 214$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3\cdot 311$ |
$3$ |
$37.1794447007$ |
$205.48177563524004$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$33.1920594291$ |
$0$ |
$311$ |
| 4.2.2201600.1 |
$x^{4} - 8 x^{2} - 344$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5^{2}\cdot 43$ |
$3$ |
$38.5198514961$ |
$98.63969438885592$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$179.928022239$ |
$1$ |
$43$ |
| 4.2.2250752.2 |
$x^{4} - 24 x^{2} - 8 x + 118$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 7\cdot 157$ |
$3$ |
$38.7330691493$ |
$223.01359277303482$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$55.9650110207$ |
$1$ |
$157$ |
| 4.2.2267136.2 |
$x^{4} - 24 x^{2} - 96 x + 168$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3^{3}\cdot 41$ |
$3$ |
$38.8033653897$ |
$98.18956985135738$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[2, 4]$ |
$2$ |
$2$ |
$34.7476829868$ |
$1$ |
$41$ |
| 4.0.2304000.1 |
$x^{4} + 60 x^{2} + 90$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 5^{3}$ |
$3$ |
$38.960149857$ |
$38.96014985701187$ |
✓ |
✓ |
|
$C_4$ (as 4T1) |
$[2, 2, 4]$ |
$[2, 2, 4]$ |
$2$ |
$1$ |
$3.63689291846$ |
$0$ |
$5$ |
| 4.2.2316288.2 |
$x^{4} - 4 x^{2} - 16 x + 26$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 13\cdot 29$ |
$4$ |
$39.0119931518$ |
$226.23708195034487$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$75.5087470534$ |
$1$ |
$29$ |
| 4.2.2332672.2 |
$x^{4} + 4 x^{2} - 48 x - 4$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 17\cdot 67$ |
$3$ |
$39.0807976398$ |
$227.03580317072144$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$34.078630034$ |
$1$ |
$67$ |
| 4.2.2463744.2 |
$x^{4} - 8 x^{2} - 24 x + 22$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 401$ |
$3$ |
$39.618579025$ |
$233.32716365034454$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$115.985296146$ |
$1$ |
$401$ |
| 4.0.2615296.4 |
$x^{4} + 12 x^{2} - 64 x + 186$ |
$4$ |
(0, 2) |
$2^{11}\cdot 1277$ |
$2$ |
$40.2142721228$ |
$240.39638727072034$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$18.3714283709$ |
$0$ |
$1277$ |
| 4.2.2627584.2 |
$x^{4} - 8 x^{2} - 24 x + 54$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 1283$ |
$2$ |
$40.2614259331$ |
$240.96047812201442$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$80.8763584765$ |
$1$ |
$1283$ |
| 4.2.2660352.2 |
$x^{4} - 28 x^{2} - 16 x + 188$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 433$ |
$3$ |
$40.386365917$ |
$242.45830437566954$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$91.0709240573$ |
$1$ |
$433$ |
| 4.2.2709504.5 |
$x^{4} + 12 x^{2} - 6$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3^{3}\cdot 7^{2}$ |
$3$ |
$40.5716293381$ |
$40.571629338126115$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[2, 4]$ |
$2$ |
$2$ |
$36.3311358549$ |
$1$ |
$7$ |
| 4.0.2713600.6 |
$x^{4} + 56 x^{2} + 424$ |
$4$ |
(0, 2) |
$2^{11}\cdot 5^{2}\cdot 53$ |
$3$ |
$40.5869538467$ |
$109.51041507054863$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
$[12]$ |
$[12]$ |
$2$ |
$1$ |
$3.63689291846$ |
$0$ |
$53$ |
| 4.0.2746368.5 |
$x^{4} - 8 x^{2} + 1192$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 149$ |
$3$ |
$40.7089295383$ |
$142.22837549583681$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$60.4026813977$ |
$0$ |
$149$ |
| 4.2.2873344.1 |
$x^{4} - 8 x^{2} - 24 x - 10$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 23\cdot 61$ |
$3$ |
$41.1715204684$ |
$251.97724519547887$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$46.7725441132$ |
$1$ |
$61$ |
| 4.2.2971648.1 |
$x^{4} - 4 x^{2} - 80 x - 358$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 1451$ |
$2$ |
$41.519234831$ |
$256.25136902679674$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$105.814642145$ |
$1$ |
$1451$ |
| 4.0.2992128.3 |
$x^{4} + 24 x^{2} - 40 x + 150$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3\cdot 487$ |
$3$ |
$41.5905862452$ |
$257.1328692875863$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$68.229431257$ |
$0$ |
$487$ |
| 4.2.3020800.1 |
$x^{4} - 24 x^{2} - 96 x - 88$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5^{2}\cdot 59$ |
$3$ |
$41.6898654183$ |
$151.09096562386276$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$135.949419555$ |
$1$ |
$59$ |
| 4.2.3020800.8 |
$x^{4} + 4 x^{2} - 80 x + 124$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5^{2}\cdot 59$ |
$3$ |
$41.6898654183$ |
$115.54296183152834$ |
|
|
|
$S_4$ (as 4T5) |
$[2, 2]$ |
$[2, 4]$ |
$2$ |
$2$ |
$19.8677695677$ |
$1$ |
$59$ |
| 4.4.3041280.1 |
$x^{4} - 24 x^{2} - 24 x + 54$ |
$4$ |
(4, 0) |
$2^{11}\cdot 3^{3}\cdot 5\cdot 11$ |
$4$ |
$41.7603472682$ |
$179.7443701985217$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$3$ |
$174.311967416$ |
$3$ |
$11$ |
| 4.0.3041280.8 |
$x^{4} - 72 x^{2} + 1320$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{3}\cdot 5\cdot 11$ |
$4$ |
$41.7603472682$ |
$113.7246931932339$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2, 4]$ |
$[2, 4]$ |
$2$ |
$1$ |
$6.76847079937$ |
$0$ |
$11$ |
| 4.0.3041280.10 |
$x^{4} + 24 x^{2} + 1320$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{3}\cdot 5\cdot 11$ |
$4$ |
$41.7603472682$ |
$113.7246931932339$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4, 8]$ |
$[4, 8]$ |
$2$ |
$1$ |
$6.36415283595$ |
$0$ |
$11$ |
| 4.0.3188736.2 |
$x^{4} + 28 x^{2} + 346$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 173$ |
$3$ |
$42.2575818383$ |
$153.25553446415032$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$58.045176879$ |
$0$ |
$173$ |
| 4.0.3221504.1 |
$x^{4} - 4 x^{2} + 26$ |
$4$ |
(0, 2) |
$2^{11}\cdot 11^{2}\cdot 13$ |
$3$ |
$42.3657275257$ |
$80.4452687323455$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$16.7527554663$ |
$0$ |
$13$ |
| 4.2.3299328.1 |
$x^{4} - 36 x^{2} - 144 x - 198$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3^{2}\cdot 179$ |
$3$ |
$42.6193059672$ |
$155.89049315407038$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[2, 4]$ |
$2$ |
$2$ |
$42.5720851404$ |
$1$ |
$179$ |
| 4.2.3348480.1 |
$x^{4} - 28 x^{2} - 48 x + 60$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 5\cdot 109$ |
$4$ |
$42.7771579503$ |
$272.0140687232194$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$206.563090103$ |
$1$ |
$109$ |
| 4.2.3430400.3 |
$x^{4} - 52 x^{2} - 134$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5^{2}\cdot 67$ |
$3$ |
$43.0364253214$ |
$123.12745180762728$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[4]$ |
$2$ |
$2$ |
$50.8832956574$ |
$1$ |
$67$ |
| 4.4.3565568.1 |
$x^{4} - 36 x^{2} - 16 x + 26$ |
$4$ |
(4, 0) |
$2^{11}\cdot 1741$ |
$2$ |
$43.4542422206$ |
$280.6931883514986$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[2, 4]$ |
$2$ |
$3$ |
$112.457991522$ |
$2$ |
$1741$ |
| 4.0.3565568.1 |
$x^{4} + 24 x^{2} - 72 x + 86$ |
$4$ |
(0, 2) |
$2^{11}\cdot 1741$ |
$2$ |
$43.4542422206$ |
$280.6931883514986$ |
|
|
|
$S_4$ (as 4T5) |
$[2, 4]$ |
$[2, 4]$ |
$2$ |
$1$ |
$12.4986311955$ |
$0$ |
$1741$ |
