| 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 $p$ |
| 4.2.55296.2 |
$x^{4} - 8 x + 6$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3^{3}$ |
$2$ |
$15.3346345019$ |
$24.23672593327708$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$6.2238346237$ |
$2$ |
$3$ |
| 4.0.92160.5 |
$x^{4} + 8 x^{2} + 40$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 5$ |
$3$ |
$17.4235086988$ |
$26.054222497305222$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$3.28134469131$ |
$0$ |
$5$ |
| 4.2.153600.2 |
$x^{4} + 4 x^{2} - 6$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 5^{2}$ |
$3$ |
$19.7969280154$ |
$26.054222497305222$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$5.96694963543$ |
$2$ |
$5$ |
| 4.0.256000.2 |
$x^{4} + 20 x^{2} + 90$ |
$4$ |
(0, 2) |
$2^{11}\cdot 5^{3}$ |
$2$ |
$22.4936530076$ |
$22.493653007613965$ |
✓ |
✓ |
|
$C_4$ (as 4T1) |
$[20]$ |
$[20]$ |
$2$ |
$1$ |
$3.63689291846$ |
$0$ |
$5$ |
| 4.0.387072.3 |
$x^{4} - 24 x^{2} + 168$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{3}\cdot 7$ |
$3$ |
$24.942957063$ |
$40.571629338126115$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$11.4427120678$ |
$0$ |
$7$ |
| 4.0.518144.3 |
$x^{4} - 12 x^{2} - 16 x + 92$ |
$4$ |
(0, 2) |
$2^{11}\cdot 11\cdot 23$ |
$3$ |
$26.8295024231$ |
$107.00221026209027$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$7.09952291622$ |
$0$ |
$23$ |
| 4.0.534528.2 |
$x^{4} + 4 x^{2} + 58$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 29$ |
$3$ |
$27.0391238424$ |
$62.74687687563977$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$21.0592339669$ |
$0$ |
$29$ |
| 4.2.563200.2 |
$x^{4} + 4 x^{2} - 32 x - 38$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5^{2}\cdot 11$ |
$3$ |
$27.3946440308$ |
$65.2392310516812$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$17.1370214028$ |
$2$ |
$11$ |
| 4.4.665600.1 |
$x^{4} - 12 x^{2} + 26$ |
$4$ |
(4, 0) |
$2^{11}\cdot 5^{2}\cdot 13$ |
$3$ |
$28.562967671$ |
$54.23618911516588$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$3$ |
$38.8402582446$ |
$4$ |
$13$ |
| 4.0.747520.3 |
$x^{4} + 20 x^{2} - 16 x + 28$ |
$4$ |
(0, 2) |
$2^{11}\cdot 5\cdot 73$ |
$3$ |
$29.403951937$ |
$128.52242764793135$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$12.1827342231$ |
$0$ |
$73$ |
| 4.0.927744.3 |
$x^{4} + 16 x^{2} - 24 x + 70$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3\cdot 151$ |
$3$ |
$31.0353779479$ |
$143.17974647330672$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$31.5459927766$ |
$0$ |
$151$ |
| 4.2.972800.1 |
$x^{4} - 16 x^{2} - 40 x - 26$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5^{2}\cdot 19$ |
$3$ |
$31.4055128994$ |
$65.56835539812027$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2, 4]$ |
$[2, 4]$ |
$2$ |
$2$ |
$10.5121596095$ |
$2$ |
$19$ |
| 4.0.976896.4 |
$x^{4} + 40 x^{2} + 424$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 53$ |
$3$ |
$31.4385192643$ |
$84.82640276089931$ |
|
|
|
$D_{4}$ (as 4T3) |
$[12]$ |
$[12]$ |
$2$ |
$1$ |
$9.67396283519$ |
$0$ |
$53$ |
| 4.2.1054720.1 |
$x^{4} - 8 x^{2} - 32 x + 40$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5\cdot 103$ |
$3$ |
$32.0467723538$ |
$152.66381204433685$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$17.7164810266$ |
$2$ |
$103$ |
| 4.2.1202176.1 |
$x^{4} - 24 x - 58$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 587$ |
$2$ |
$33.1125032086$ |
$162.98646433252125$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$30.2114334623$ |
$2$ |
$587$ |
| 4.0.1419264.3 |
$x^{4} + 4 x^{2} + 154$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 7\cdot 11$ |
$4$ |
$34.5156293585$ |
$102.24415217048806$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$14.7003376297$ |
$0$ |
$11$ |
| 4.0.1419264.4 |
$x^{4} + 20 x^{2} + 154$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 7\cdot 11$ |
$4$ |
$34.5156293585$ |
$102.24415217048806$ |
|
|
|
$D_{4}$ (as 4T3) |
$[12]$ |
$[12]$ |
$2$ |
$1$ |
$7.99939682902$ |
$0$ |
$11$ |
| 4.4.1583104.1 |
$x^{4} - 28 x^{2} - 64 x - 38$ |
$4$ |
(4, 0) |
$2^{11}\cdot 773$ |
$2$ |
$35.4713209239$ |
$187.03472051696946$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$3$ |
$28.7882942726$ |
$4$ |
$773$ |
| 4.2.1611776.2 |
$x^{4} + 16 x^{2} - 72 x - 26$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 787$ |
$2$ |
$35.6308490383$ |
$188.72083709756066$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$22.5765416764$ |
$2$ |
$787$ |
| 4.0.1714176.1 |
$x^{4} - 12 x^{2} + 186$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{3}\cdot 31$ |
$3$ |
$36.1837731989$ |
$85.3796314967612$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$41.9454580855$ |
$0$ |
$31$ |
| 4.4.1730560.2 |
$x^{4} - 24 x^{2} + 40$ |
$4$ |
(4, 0) |
$2^{11}\cdot 5\cdot 13^{2}$ |
$3$ |
$36.2699256496$ |
$54.23618911516588$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$3$ |
$49.3914472398$ |
$4$ |
$13$ |
| 4.0.1861632.4 |
$x^{4} - 56 x^{2} + 808$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 101$ |
$3$ |
$36.9380069676$ |
$117.09916609766924$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$40.7799178938$ |
$0$ |
$101$ |
| 4.0.1894400.6 |
$x^{4} + 52 x^{2} + 666$ |
$4$ |
(0, 2) |
$2^{11}\cdot 5^{2}\cdot 37$ |
$3$ |
$37.0994884429$ |
$91.49942234379802$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
$[28]$ |
$[28]$ |
$2$ |
$1$ |
$3.63689291846$ |
$0$ |
$37$ |
| 4.4.1910784.1 |
$x^{4} - 16 x^{2} - 8 x + 38$ |
$4$ |
(4, 0) |
$2^{11}\cdot 3\cdot 311$ |
$3$ |
$37.1794447007$ |
$205.48177563524004$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$3$ |
$47.0009983531$ |
$4$ |
$311$ |
| 4.2.1923072.1 |
$x^{4} + 20 x^{2} - 16 x - 100$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 313$ |
$3$ |
$37.2390751164$ |
$206.1414299023531$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$45.4643411355$ |
$2$ |
$313$ |
| 4.0.1927168.3 |
$x^{4} - 16 x^{2} - 40 x + 262$ |
$4$ |
(0, 2) |
$2^{11}\cdot 941$ |
$2$ |
$37.2588884115$ |
$206.3608460686732$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$19.0745373993$ |
$0$ |
$941$ |
| 4.2.1972224.1 |
$x^{4} - 4 x^{2} - 96 x - 92$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3^{2}\cdot 107$ |
$3$ |
$37.4747772306$ |
$120.52718246394227$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[2, 4]$ |
$2$ |
$2$ |
$14.9858733313$ |
$1$ |
$107$ |
| 4.2.2037760.2 |
$x^{4} - 4 x^{2} - 96 x + 164$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5\cdot 199$ |
$3$ |
$37.7822883569$ |
$212.19933983393855$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$22.182867004$ |
$2$ |
$199$ |
| 4.2.2136064.1 |
$x^{4} + 36 x^{2} - 16 x - 70$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 7\cdot 149$ |
$3$ |
$38.2299354901$ |
$217.25743222675817$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$24.9396086879$ |
$2$ |
$149$ |
| 4.2.2201600.2 |
$x^{4} + 8 x^{2} - 344$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5^{2}\cdot 43$ |
$3$ |
$38.5198514961$ |
$98.63969438885592$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$28.0839898657$ |
$2$ |
$43$ |
| 4.2.2250752.1 |
$x^{4} + 4 x^{2} - 16 x - 70$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 7\cdot 157$ |
$3$ |
$38.7330691493$ |
$223.01359277303482$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$46.2459332997$ |
$2$ |
$157$ |
| 4.4.2304000.1 |
$x^{4} - 60 x^{2} + 90$ |
$4$ |
(4, 0) |
$2^{11}\cdot 3^{2}\cdot 5^{3}$ |
$3$ |
$38.960149857$ |
$38.96014985701187$ |
|
✓ |
✓ |
$C_4$ (as 4T1) |
$[4]$ |
$[2, 2, 4]$ |
$2$ |
$3$ |
$52.3627778724$ |
$2$ |
$5$ |
| 4.2.2316288.1 |
$x^{4} - 16 x^{2} - 24 x + 6$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 13\cdot 29$ |
$4$ |
$39.0119931518$ |
$226.23708195034487$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$69.4227118613$ |
$2$ |
$29$ |
| 4.2.2594816.1 |
$x^{4} - 8 x^{2} - 32 x - 88$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 7\cdot 181$ |
$3$ |
$40.1353118516$ |
$239.4532828608845$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$42.3956874311$ |
$2$ |
$181$ |
| 4.2.2611200.2 |
$x^{4} + 4 x^{2} - 48 x - 38$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3\cdot 5^{2}\cdot 17$ |
$4$ |
$40.1985172629$ |
$140.47452718907857$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$39.545267571$ |
$2$ |
$17$ |
| 4.0.2615296.3 |
$x^{4} + 24 x^{2} - 32 x + 40$ |
$4$ |
(0, 2) |
$2^{11}\cdot 1277$ |
$2$ |
$40.2142721228$ |
$240.39638727072034$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$25.643801505$ |
$0$ |
$1277$ |
| 4.2.2709504.4 |
$x^{4} - 12 x^{2} - 6$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3^{3}\cdot 7^{2}$ |
$3$ |
$40.5716293381$ |
$40.571629338126115$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[2, 4]$ |
$2$ |
$2$ |
$37.2644958007$ |
$1$ |
$7$ |
| 4.2.2709504.10 |
$x^{4} - 24 x^{2} - 64 x - 216$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 3^{3}\cdot 7^{2}$ |
$3$ |
$40.5716293381$ |
$88.68958959990265$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$45.7314468946$ |
$2$ |
$7$ |
| 4.4.2713600.2 |
$x^{4} - 56 x^{2} + 424$ |
$4$ |
(4, 0) |
$2^{11}\cdot 5^{2}\cdot 53$ |
$3$ |
$40.5869538467$ |
$109.51041507054863$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$3$ |
$107.284126329$ |
$4$ |
$53$ |
| 4.0.2746368.6 |
$x^{4} + 8 x^{2} + 1192$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 149$ |
$3$ |
$40.7089295383$ |
$142.22837549583681$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$14.2503183587$ |
$0$ |
$149$ |
| 4.2.2824192.1 |
$x^{4} + 4 x^{2} - 112 x - 518$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 7\cdot 197$ |
$3$ |
$40.9943075159$ |
$249.81276204469606$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$32.7120684175$ |
$2$ |
$197$ |
| 4.2.2873344.2 |
$x^{4} + 4 x^{2} - 16 x - 6$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 23\cdot 61$ |
$3$ |
$41.1715204684$ |
$251.97724519547887$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$56.3387473963$ |
$2$ |
$61$ |
| 4.4.2992128.1 |
$x^{4} - 28 x^{2} - 32 x + 26$ |
$4$ |
(4, 0) |
$2^{11}\cdot 3\cdot 487$ |
$3$ |
$41.5905862452$ |
$257.1328692875863$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$3$ |
$61.897344661$ |
$4$ |
$487$ |
| 4.2.3020800.2 |
$x^{4} - 24 x^{2} - 64 x - 88$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5^{2}\cdot 59$ |
$3$ |
$41.6898654183$ |
$151.09096562386276$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$25.8534434429$ |
$2$ |
$59$ |
| 4.2.3020800.6 |
$x^{4} - 44 x^{2} - 144 x - 38$ |
$4$ |
(2, 1) |
$-\,2^{11}\cdot 5^{2}\cdot 59$ |
$3$ |
$41.6898654183$ |
$151.09096562386276$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$2$ |
$63.1516372833$ |
$2$ |
$59$ |
| 4.0.3041280.3 |
$x^{4} - 12 x^{2} - 16 x + 90$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{3}\cdot 5\cdot 11$ |
$4$ |
$41.7603472682$ |
$179.7443701985217$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$53.2070090477$ |
$0$ |
$11$ |
| 4.0.3041280.9 |
$x^{4} - 24 x^{2} + 1320$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{3}\cdot 5\cdot 11$ |
$4$ |
$41.7603472682$ |
$113.7246931932339$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2, 4]$ |
$[2, 4]$ |
$2$ |
$1$ |
$13.1314778119$ |
$0$ |
$11$ |
| 4.0.3041280.11 |
$x^{4} + 72 x^{2} + 1320$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{3}\cdot 5\cdot 11$ |
$4$ |
$41.7603472682$ |
$113.7246931932339$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2, 4]$ |
$[2, 4]$ |
$2$ |
$1$ |
$18.770851004$ |
$0$ |
$11$ |
| 4.0.3188736.1 |
$x^{4} - 28 x^{2} + 346$ |
$4$ |
(0, 2) |
$2^{11}\cdot 3^{2}\cdot 173$ |
$3$ |
$42.2575818383$ |
$153.25553446415032$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$42.2351294991$ |
$0$ |
$173$ |
| 4.0.3221504.2 |
$x^{4} + 4 x^{2} + 26$ |
$4$ |
(0, 2) |
$2^{11}\cdot 11^{2}\cdot 13$ |
$3$ |
$42.3657275257$ |
$80.4452687323455$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$15.0582016221$ |
$0$ |
$13$ |
