| 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 $p$ |
| 4.0.189.1 |
$x^{4} - x^{3} + 2 x + 1$ |
$4$ |
(0, 2) |
$3^{3}\cdot 7$ |
$2$ |
$3.70779275107$ |
$6.03100878451909$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$0.862554627662$ |
$0$ |
$7$ |
| 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 |
trivial |
$2$ |
$2$ |
$0.557886846205$ |
$2$ |
$7$ |
| 4.0.1008.1 |
$x^{4} - 5 x^{2} + 7$ |
$4$ |
(0, 2) |
$2^{4}\cdot 3^{2}\cdot 7$ |
$3$ |
$5.63462649452$ |
$12.96148139681572$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$3.39588318368$ |
$0$ |
$7$ |
| 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 |
trivial |
$2$ |
$2$ |
$1.76052319498$ |
$2$ |
$13$ |
| 4.0.1953.1 |
$x^{4} - 2 x^{3} - 5 x^{2} + 6 x + 12$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 31$ |
$3$ |
$6.64776350271$ |
$25.514701644346147$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$5.73065616586$ |
$0$ |
$31$ |
| 4.0.1953.2 |
$x^{4} - x^{3} + 8 x^{2} - 5 x + 13$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 31$ |
$3$ |
$6.64776350271$ |
$25.514701644346147$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$3.77444756158$ |
$0$ |
$31$ |
| 4.2.2051.1 |
$x^{4} - x^{3} - x^{2} + 3 x + 1$ |
$4$ |
(2, 1) |
$-\,7\cdot 293$ |
$2$ |
$6.72963353363$ |
$45.28796749689701$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$1.74300316753$ |
$2$ |
$293$ |
| 4.2.2191.1 |
$x^{4} - x^{3} + 3 x^{2} - 1$ |
$4$ |
(2, 1) |
$-\,7\cdot 313$ |
$2$ |
$6.84164592433$ |
$46.8081189538738$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$2.32583009081$ |
$2$ |
$313$ |
| 4.2.2219.1 |
$x^{4} - x^{3} + 3 x^{2} - x - 1$ |
$4$ |
(2, 1) |
$-\,7\cdot 317$ |
$2$ |
$6.86340023991$ |
$47.10626285325551$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$1.50383646997$ |
$2$ |
$317$ |
| 4.0.2296.1 |
$x^{4} - x^{3} - 2 x^{2} + x + 3$ |
$4$ |
(0, 2) |
$2^{3}\cdot 7\cdot 41$ |
$3$ |
$6.92218131826$ |
$47.916594202843754$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$1$ |
$2.58271206295$ |
$0$ |
$41$ |
| 4.0.2429.1 |
$x^{4} - x^{3} - 2 x^{2} + 2 x + 3$ |
$4$ |
(0, 2) |
$7\cdot 347$ |
$2$ |
$7.02031951705$ |
$49.28488612140643$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$1$ |
$1.89610142329$ |
$0$ |
$347$ |
| 4.2.2443.1 |
$x^{4} - 3 x - 1$ |
$4$ |
(2, 1) |
$-\,7\cdot 349$ |
$2$ |
$7.03041346046$ |
$49.42671342502959$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$2.221501113$ |
$2$ |
$349$ |
| 4.0.2457.2 |
$x^{4} - x^{3} - 6 x^{2} - x + 19$ |
$4$ |
(0, 2) |
$3^{3}\cdot 7\cdot 13$ |
$3$ |
$7.04046411294$ |
$21.745111415357325$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$6$ |
$1$ |
$2.68739144281$ |
$0$ |
$13$ |
| 4.0.2709.1 |
$x^{4} - x^{3} - 7 x^{2} + 7 x + 19$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 43$ |
$3$ |
$7.21443377672$ |
$30.04995840263344$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$2.95085943144$ |
$0$ |
$43$ |
| 4.0.3024.1 |
$x^{4} - 2 x^{3} + 3$ |
$4$ |
(0, 2) |
$2^{4}\cdot 3^{3}\cdot 7$ |
$3$ |
$7.41558550213$ |
$21.398934305093306$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$1$ |
$3.35716994393$ |
$0$ |
$7$ |
| 4.2.3052.1 |
$x^{4} - x^{3} + x^{2} - x - 2$ |
$4$ |
(2, 1) |
$-\,2^{2}\cdot 7\cdot 109$ |
$3$ |
$7.43269192606$ |
$43.847913543743175$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$4.08318775963$ |
$2$ |
$109$ |
| 4.2.4032.1 |
$x^{4} - 2 x^{3} - x^{2} - 4 x - 2$ |
$4$ |
(2, 1) |
$-\,2^{6}\cdot 3^{2}\cdot 7$ |
$3$ |
$7.96856520746$ |
$12.96148139681572$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$4.93352004952$ |
$1$ |
$7$ |
| 4.0.4032.2 |
$x^{4} - 10 x^{2} + 28$ |
$4$ |
(0, 2) |
$2^{6}\cdot 3^{2}\cdot 7$ |
$3$ |
$7.96856520746$ |
$18.33030277982336$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$4.43462099913$ |
$0$ |
$7$ |
| 4.0.4221.2 |
$x^{4} - 2 x^{3} - 8 x^{2} + 9 x + 27$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 67$ |
$3$ |
$8.06034884376$ |
$37.5099986670221$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$6.94481290844$ |
$0$ |
$67$ |
| 4.2.4319.1 |
$x^{4} - x^{3} - 4 x^{2} - x + 2$ |
$4$ |
(2, 1) |
$-\,7\cdot 617$ |
$2$ |
$8.10673172139$ |
$65.71909920259102$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$4.0888753021$ |
$2$ |
$617$ |
| 4.2.4564.1 |
$x^{4} - x^{3} - 5 x + 1$ |
$4$ |
(2, 1) |
$-\,2^{2}\cdot 7\cdot 163$ |
$3$ |
$8.21932983985$ |
$67.55738301621814$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$4.06824093953$ |
$2$ |
$163$ |
| 4.2.4648.1 |
$x^{4} + x^{2} - 6 x - 4$ |
$4$ |
(2, 1) |
$-\,2^{3}\cdot 7\cdot 83$ |
$3$ |
$8.25689058885$ |
$68.17624219623724$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$3.84770233495$ |
$2$ |
$83$ |
| 4.2.4907.1 |
$x^{4} - x^{3} - 2 x^{2} - 4 x - 1$ |
$4$ |
(2, 1) |
$-\,7\cdot 701$ |
$2$ |
$8.36958673744$ |
$70.04998215560087$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$2.67065772309$ |
$2$ |
$701$ |
| 4.0.4977.2 |
$x^{4} - x^{3} - 11 x^{2} + 6 x + 36$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 79$ |
$3$ |
$8.39927711758$ |
$40.7308237088326$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$10.2452243803$ |
$0$ |
$79$ |
| 4.2.5348.1 |
$x^{4} - x^{3} + 4 x^{2} - x - 1$ |
$4$ |
(2, 1) |
$-\,2^{2}\cdot 7\cdot 191$ |
$3$ |
$8.55160927517$ |
$73.13002119512889$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$5.12711239714$ |
$2$ |
$191$ |
| 4.2.5432.1 |
$x^{4} - x^{3} + x - 3$ |
$4$ |
(2, 1) |
$-\,2^{3}\cdot 7\cdot 97$ |
$3$ |
$8.58499289987$ |
$73.70210309075311$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$5.14834890155$ |
$2$ |
$97$ |
| 4.2.5887.1 |
$x^{4} - x^{3} - 3 x^{2} - 2 x + 4$ |
$4$ |
(2, 1) |
$-\,7\cdot 29^{2}$ |
$2$ |
$8.75938285616$ |
$14.247806848775006$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$2$ |
$2$ |
$3.66933704259$ |
$2$ |
$29$ |
| 4.0.6209.1 |
$x^{4} + x^{2} - x + 3$ |
$4$ |
(0, 2) |
$7\cdot 887$ |
$2$ |
$8.87677914969$ |
$78.79720807236764$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$1$ |
$5.47370730723$ |
$0$ |
$887$ |
| 4.0.6489.1 |
$x^{4} - x^{3} + 14 x^{2} - 5 x + 43$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 103$ |
$3$ |
$8.97520637604$ |
$46.50806381693394$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$6.95569001881$ |
$0$ |
$103$ |
| 4.0.6489.2 |
$x^{4} - x^{3} - 11 x^{2} + 48$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 103$ |
$3$ |
$8.97520637604$ |
$46.50806381693394$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$11.971170067$ |
$0$ |
$103$ |
| 4.0.6685.1 |
$x^{4} - 2 x^{3} + 4 x^{2} - x + 3$ |
$4$ |
(0, 2) |
$5\cdot 7\cdot 191$ |
$3$ |
$9.04222590373$ |
$81.7618492941543$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$1$ |
$2.49243428852$ |
$0$ |
$191$ |
| 4.0.6832.1 |
$x^{4} - x^{2} - 2 x + 9$ |
$4$ |
(0, 2) |
$2^{4}\cdot 7\cdot 61$ |
$3$ |
$9.09152975462$ |
$58.44655678480983$ |
|
|
|
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$1$ |
$3.3284420423$ |
$0$ |
$61$ |
| 4.4.7168.1 |
$x^{4} - 6 x^{2} + 7$ |
$4$ |
(4, 0) |
$2^{10}\cdot 7$ |
$2$ |
$9.20130653516$ |
$17.798422345016238$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$3$ |
$5.71497174049$ |
$3$ |
$7$ |
| 4.2.7259.1 |
$x^{4} + 4 x^{2} - x - 1$ |
$4$ |
(2, 1) |
$-\,7\cdot 17\cdot 61$ |
$3$ |
$9.23037189164$ |
$85.19976525789258$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$3.59922058717$ |
$2$ |
$61$ |
| 4.0.7504.1 |
$x^{4} + 5 x^{2} - 2 x + 1$ |
$4$ |
(0, 2) |
$2^{4}\cdot 7\cdot 67$ |
$3$ |
$9.30728914942$ |
$43.31281565541543$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$1.60432887661$ |
$0$ |
$67$ |
| 4.2.7532.1 |
$x^{4} - 4 x^{2} - 2 x - 3$ |
$4$ |
(2, 1) |
$-\,2^{2}\cdot 7\cdot 269$ |
$3$ |
$9.31595919994$ |
$68.88296359698886$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$8.7931660691$ |
$2$ |
$269$ |
| 4.2.7616.2 |
$x^{4} - 2 x^{3} + 5 x^{2} + 6 x - 9$ |
$4$ |
(2, 1) |
$-\,2^{6}\cdot 7\cdot 17$ |
$3$ |
$9.34182513843$ |
$30.854497241083024$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$3.05293335278$ |
$1$ |
$17$ |
| 4.0.8001.2 |
$x^{4} - x^{3} - 14 x^{2} + 9 x + 57$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 127$ |
$3$ |
$9.45771162043$ |
$51.64300533470143$ |
|
|
|
$D_{4}$ (as 4T3) |
$[3]$ |
$[3]$ |
$6$ |
$1$ |
$4.50217431324$ |
$0$ |
$127$ |
| 4.0.8057.1 |
$x^{4} - 2 x^{3} + 5 x^{2} - x + 2$ |
$4$ |
(0, 2) |
$7\cdot 1151$ |
$2$ |
$9.47421728814$ |
$89.76079322287654$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$1$ |
$4.27431774552$ |
$0$ |
$1151$ |
| 4.0.8092.1 |
$x^{4} - x^{3} + 5 x^{2} - 4 x + 6$ |
$4$ |
(0, 2) |
$2^{2}\cdot 7\cdot 17^{2}$ |
$3$ |
$9.48448968454$ |
$34.984711477348505$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$1$ |
$5.04437571067$ |
$0$ |
$17$ |
| 4.2.8127.1 |
$x^{4} - x^{3} - 7 x - 2$ |
$4$ |
(2, 1) |
$-\,3^{3}\cdot 7\cdot 43$ |
$3$ |
$9.49472881171$ |
$62.50643235670045$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$7.39292359987$ |
$2$ |
$43$ |
| 4.0.8757.1 |
$x^{4} - 2 x^{3} - 14 x^{2} + 15 x + 57$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 139$ |
$3$ |
$9.67361585773$ |
$54.0277706369604$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$9.87643489185$ |
$0$ |
$139$ |
| 4.0.8757.2 |
$x^{4} - x^{3} - 13 x^{2} + 13 x + 61$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 139$ |
$3$ |
$9.67361585773$ |
$54.0277706369604$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$4.55680865694$ |
$0$ |
$139$ |
| 4.2.8960.1 |
$x^{4} - 4 x - 2$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 5\cdot 7$ |
$3$ |
$9.72919711639$ |
$26.562300125198664$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$6.77162446352$ |
$2$ |
$7$ |
| 4.2.9247.1 |
$x^{4} - x^{3} - 3 x^{2} + 6 x - 1$ |
$4$ |
(2, 1) |
$-\,7\cdot 1321$ |
$2$ |
$9.80618798444$ |
$96.1613227862429$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$5.01419715012$ |
$2$ |
$1321$ |
| 4.0.9513.1 |
$x^{4} - x^{3} + 17 x^{2} - 8 x + 64$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 151$ |
$3$ |
$9.87596118067$ |
$56.31163290120435$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$8.86177553195$ |
$0$ |
$151$ |
| 4.2.9583.1 |
$x^{4} - 3 x^{2} - 7$ |
$4$ |
(2, 1) |
$-\,7\cdot 37^{2}$ |
$2$ |
$9.89407896216$ |
$16.09347693943108$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$2$ |
$2$ |
$6.57797666237$ |
$2$ |
$37$ |
| 4.2.9667.1 |
$x^{4} - 2 x^{3} - 2 x^{2} - x + 2$ |
$4$ |
(2, 1) |
$-\,7\cdot 1381$ |
$2$ |
$9.91568974752$ |
$98.32090316916337$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$9.48334045014$ |
$2$ |
$1381$ |
| 4.2.10059.1 |
$x^{4} - x^{3} - 2 x^{2} - 6 x + 3$ |
$4$ |
(2, 1) |
$-\,3\cdot 7\cdot 479$ |
$3$ |
$10.0147174775$ |
$100.29456615390487$ |
|
|
|
$S_4$ (as 4T5) |
trivial |
trivial |
$2$ |
$2$ |
$4.38511200459$ |
$2$ |
$479$ |
| 4.0.10269.2 |
$x^{4} - x^{3} - 14 x^{2} + 75$ |
$4$ |
(0, 2) |
$3^{2}\cdot 7\cdot 163$ |
$3$ |
$10.0665820676$ |
$58.5064099052403$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
trivial |
$6$ |
$1$ |
$10.2278465093$ |
$0$ |
$163$ |
