| 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 $p$ |
| 4.2.6912.1 |
$x^{4} - 3$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{3}$ |
$2$ |
$9.11802822782$ |
$9.11802822781911$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$5.24584468864$ |
$1$ |
$3$ |
| 4.2.21248.1 |
$x^{4} + 2 x^{2} - 4 x - 2$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 83$ |
$2$ |
$12.0733979172$ |
$36.4417343165772$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$12.2969025794$ |
$1$ |
$83$ |
| 4.2.25344.1 |
$x^{4} - 8 x^{2} - 11$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{2}\cdot 11$ |
$3$ |
$12.6173685821$ |
$22.978250586152114$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$13.757446716$ |
$1$ |
$11$ |
| 4.2.25344.2 |
$x^{4} + 8 x^{2} - 11$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{2}\cdot 11$ |
$3$ |
$12.6173685821$ |
$22.978250586152114$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$8.18283102253$ |
$1$ |
$11$ |
| 4.2.27392.1 |
$x^{4} - 6 x^{2} - 4 x - 2$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 107$ |
$2$ |
$12.8648858108$ |
$41.376321731154405$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$10.5877098931$ |
$1$ |
$107$ |
| 4.0.29952.1 |
$x^{4} + 8 x^{2} + 13$ |
$4$ |
(0, 2) |
$2^{8}\cdot 3^{2}\cdot 13$ |
$3$ |
$13.1554726719$ |
$24.979991993593593$ |
✓ |
|
✓ |
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$2.63391579385$ |
$0$ |
$13$ |
| 4.4.29952.1 |
$x^{4} - 8 x^{2} + 13$ |
$4$ |
(4, 0) |
$2^{8}\cdot 3^{2}\cdot 13$ |
$3$ |
$13.1554726719$ |
$24.979991993593593$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$3$ |
$20.1489239619$ |
$3$ |
$13$ |
| 4.0.32000.1 |
$x^{4} + 5$ |
$4$ |
(0, 2) |
$2^{8}\cdot 5^{3}$ |
$2$ |
$13.3748060995$ |
$13.37480609952844$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$3.57964082014$ |
$0$ |
$5$ |
| 4.2.35584.1 |
$x^{4} - 8 x^{2} - 8 x - 3$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 139$ |
$2$ |
$13.7345264921$ |
$47.15930449020638$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$20.1005549715$ |
$1$ |
$139$ |
| 4.2.54016.2 |
$x^{4} - 8 x^{2} - 16 x - 11$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 211$ |
$2$ |
$15.2451115031$ |
$58.1033561853358$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$12.2459982572$ |
$1$ |
$211$ |
| 4.0.58624.1 |
$x^{4} + 2 x^{2} - 4 x + 6$ |
$4$ |
(0, 2) |
$2^{8}\cdot 229$ |
$2$ |
$15.5603321046$ |
$60.530983801686226$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$6.27628813547$ |
$0$ |
$229$ |
| 4.2.62208.1 |
$x^{4} - 6 x^{2} - 4 x + 6$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{5}$ |
$2$ |
$15.7928881554$ |
$29.976594395601754$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$29.4028401705$ |
$1$ |
$3$ |
| 4.2.78592.1 |
$x^{4} - 8 x + 5$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 307$ |
$2$ |
$16.7434359522$ |
$70.08566187174092$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$18.5954756893$ |
$1$ |
$307$ |
| 4.0.85248.1 |
$x^{4} + 16 x^{2} + 37$ |
$4$ |
(0, 2) |
$2^{8}\cdot 3^{2}\cdot 37$ |
$3$ |
$17.0872057825$ |
$42.14261501141095$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
$[6]$ |
$[6]$ |
$2$ |
$1$ |
$2.63391579385$ |
$0$ |
$37$ |
| 4.4.85248.1 |
$x^{4} - 16 x^{2} + 37$ |
$4$ |
(4, 0) |
$2^{8}\cdot 3^{2}\cdot 37$ |
$3$ |
$17.0872057825$ |
$42.14261501141095$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$3$ |
$23.5757434971$ |
$3$ |
$37$ |
| 4.2.97024.1 |
$x^{4} - 6 x^{2} - 4 x + 14$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 379$ |
$2$ |
$17.6489874311$ |
$77.87168933572714$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$19.3499974137$ |
$1$ |
$379$ |
| 4.2.105216.1 |
$x^{4} - 10 x^{2} - 16 x + 17$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3\cdot 137$ |
$3$ |
$18.0102792572$ |
$81.09253973085318$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$28.4504791872$ |
$1$ |
$137$ |
| 4.2.115456.1 |
$x^{4} - 6 x^{2} - 12 x - 10$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 11\cdot 41$ |
$3$ |
$18.4333439534$ |
$84.94704232638121$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$39.3821276622$ |
$1$ |
$41$ |
| 4.2.117504.1 |
$x^{4} - 8 x - 3$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{3}\cdot 17$ |
$3$ |
$18.5145500774$ |
$59.41907899077985$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$35.2306513667$ |
$1$ |
$17$ |
| 4.4.120064.1 |
$x^{4} - 14 x^{2} - 4 x + 38$ |
$4$ |
(4, 0) |
$2^{8}\cdot 7\cdot 67$ |
$3$ |
$18.6145782988$ |
$86.62563131083085$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$3$ |
$39.2622317653$ |
$3$ |
$67$ |
| 4.0.134400.1 |
$x^{4} - 8 x^{2} + 21$ |
$4$ |
(0, 2) |
$2^{8}\cdot 3\cdot 5^{2}\cdot 7$ |
$4$ |
$19.1469594348$ |
$40.98780306383839$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$7.15739260477$ |
$0$ |
$7$ |
| 4.0.134400.2 |
$x^{4} + 8 x^{2} + 21$ |
$4$ |
(0, 2) |
$2^{8}\cdot 3\cdot 5^{2}\cdot 7$ |
$4$ |
$19.1469594348$ |
$40.98780306383839$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$10.6110711535$ |
$0$ |
$7$ |
| 4.0.134400.3 |
$x^{4} + 4 x^{2} + 84$ |
$4$ |
(0, 2) |
$2^{8}\cdot 3\cdot 5^{2}\cdot 7$ |
$4$ |
$19.1469594348$ |
$40.98780306383839$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$3.5918936795$ |
$0$ |
$7$ |
| 4.0.134400.4 |
$x^{4} - 4 x^{2} + 84$ |
$4$ |
(0, 2) |
$2^{8}\cdot 3\cdot 5^{2}\cdot 7$ |
$4$ |
$19.1469594348$ |
$40.98780306383839$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$9.44509263114$ |
$0$ |
$7$ |
| 4.2.135936.1 |
$x^{4} - 8 x^{2} - 59$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{2}\cdot 59$ |
$3$ |
$19.2014321314$ |
$53.21653878260028$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$34.865204748$ |
$1$ |
$59$ |
| 4.2.135936.2 |
$x^{4} + 8 x^{2} - 59$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{2}\cdot 59$ |
$3$ |
$19.2014321314$ |
$53.21653878260028$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$18.3529024793$ |
$1$ |
$59$ |
| 4.0.140544.1 |
$x^{4} + 16 x^{2} + 61$ |
$4$ |
(0, 2) |
$2^{8}\cdot 3^{2}\cdot 61$ |
$3$ |
$19.3621275805$ |
$54.11099703387473$ |
✓ |
|
✓ |
$D_{4}$ (as 4T3) |
$[10]$ |
$[10]$ |
$2$ |
$1$ |
$2.63391579385$ |
$0$ |
$61$ |
| 4.4.140544.1 |
$x^{4} - 16 x^{2} + 61$ |
$4$ |
(4, 0) |
$2^{8}\cdot 3^{2}\cdot 61$ |
$3$ |
$19.3621275805$ |
$54.11099703387473$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$3$ |
$30.5354054605$ |
$3$ |
$61$ |
| 4.0.154880.1 |
$x^{4} + 8 x^{2} + 5$ |
$4$ |
(0, 2) |
$2^{8}\cdot 5\cdot 11^{2}$ |
$3$ |
$19.8380433521$ |
$29.664793948382652$ |
✓ |
|
✓ |
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$5.98644569225$ |
$0$ |
$11$ |
| 4.4.154880.1 |
$x^{4} - 8 x^{2} + 5$ |
$4$ |
(4, 0) |
$2^{8}\cdot 5\cdot 11^{2}$ |
$3$ |
$19.8380433521$ |
$29.664793948382652$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$3$ |
$38.5087831685$ |
$3$ |
$11$ |
| 4.0.158976.1 |
$x^{4} + 6 x^{2} - 16 x + 33$ |
$4$ |
(0, 2) |
$2^{8}\cdot 3^{3}\cdot 23$ |
$3$ |
$19.967922912$ |
$69.11389568573122$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$17.1354707202$ |
$0$ |
$23$ |
| 4.0.158976.2 |
$x^{4} + 24 x^{2} + 69$ |
$4$ |
(0, 2) |
$2^{8}\cdot 3^{3}\cdot 23$ |
$3$ |
$19.967922912$ |
$43.728527205430105$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
$[2, 2]$ |
$[2, 2]$ |
$2$ |
$1$ |
$2.63391579385$ |
$0$ |
$23$ |
| 4.4.158976.2 |
$x^{4} - 24 x^{2} + 69$ |
$4$ |
(4, 0) |
$2^{8}\cdot 3^{3}\cdot 23$ |
$3$ |
$19.967922912$ |
$43.728527205430105$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2, 2]$ |
$2$ |
$3$ |
$40.8454543817$ |
$2$ |
$23$ |
| 4.2.172800.1 |
$x^{4} - 6 x^{2} - 12 x - 18$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{3}\cdot 5^{2}$ |
$3$ |
$20.3885309382$ |
$42.138731512919776$ |
|
|
|
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$54.2287906714$ |
$1$ |
$5$ |
| 4.2.172800.5 |
$x^{4} - 75$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{3}\cdot 5^{2}$ |
$3$ |
$20.3885309382$ |
$20.38853093816547$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[2, 2]$ |
$2$ |
$2$ |
$12.2317161855$ |
$1$ |
$5$ |
| 4.2.180992.1 |
$x^{4} - 10 x^{2} - 16 x + 49$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 7\cdot 101$ |
$3$ |
$20.6259919909$ |
$106.35788640246665$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$36.7371443316$ |
$1$ |
$101$ |
| 4.0.185600.1 |
$x^{4} - 12 x^{2} + 116$ |
$4$ |
(0, 2) |
$2^{8}\cdot 5^{2}\cdot 29$ |
$3$ |
$20.7560397131$ |
$48.16637831516918$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$12.4533905058$ |
$0$ |
$29$ |
| 4.0.185600.2 |
$x^{4} + 12 x^{2} + 116$ |
$4$ |
(0, 2) |
$2^{8}\cdot 5^{2}\cdot 29$ |
$3$ |
$20.7560397131$ |
$48.16637831516918$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$8.76710455775$ |
$0$ |
$29$ |
| 4.0.187648.1 |
$x^{4} + 14 x^{2} - 16 x + 25$ |
$4$ |
(0, 2) |
$2^{8}\cdot 733$ |
$2$ |
$20.8130623374$ |
$108.29589096544707$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$8.25407456978$ |
$0$ |
$733$ |
| 4.0.187648.2 |
$x^{4} - 2 x^{2} - 16 x + 41$ |
$4$ |
(0, 2) |
$2^{8}\cdot 733$ |
$2$ |
$20.8130623374$ |
$108.29589096544707$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$4.71750754425$ |
$0$ |
$733$ |
| 4.2.191232.1 |
$x^{4} - 16 x^{2} - 83$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{2}\cdot 83$ |
$3$ |
$20.9117386125$ |
$63.118935352238$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$15.8356206425$ |
$1$ |
$83$ |
| 4.2.191232.2 |
$x^{4} + 16 x^{2} - 83$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{2}\cdot 83$ |
$3$ |
$20.9117386125$ |
$63.118935352238$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$46.1338536042$ |
$1$ |
$83$ |
| 4.2.193280.1 |
$x^{4} - 8 x^{2} - 8 x + 5$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 5\cdot 151$ |
$3$ |
$20.9675037439$ |
$109.90905331227269$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$45.0840905555$ |
$1$ |
$151$ |
| 4.2.197376.1 |
$x^{4} - 16 x - 3$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3\cdot 257$ |
$3$ |
$21.0777178096$ |
$111.06754701531857$ |
|
|
|
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$71.3654612928$ |
$1$ |
$257$ |
| 4.2.215808.2 |
$x^{4} - 16 x^{2} - 32 x - 19$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3\cdot 281$ |
$3$ |
$21.5534543986$ |
$116.1378491276638$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$32.6822693263$ |
$1$ |
$281$ |
| 4.2.226048.1 |
$x^{4} - 14 x^{2} - 28 x - 18$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 883$ |
$2$ |
$21.8047025813$ |
$118.86126366482901$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$53.2034669234$ |
$1$ |
$883$ |
| 4.2.246528.1 |
$x^{4} - 32 x^{2} - 107$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{2}\cdot 107$ |
$3$ |
$22.2826358579$ |
$71.66589146867567$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$49.534098085$ |
$1$ |
$107$ |
| 4.2.246528.2 |
$x^{4} + 32 x^{2} - 107$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 3^{2}\cdot 107$ |
$3$ |
$22.2826358579$ |
$71.66589146867567$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$14.3270022162$ |
$1$ |
$107$ |
| 4.2.248576.1 |
$x^{4} - 6 x^{2} - 12 x - 2$ |
$4$ |
(2, 1) |
$-\,2^{8}\cdot 971$ |
$2$ |
$22.3287699264$ |
$124.64349160706307$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$35.6094906664$ |
$1$ |
$971$ |
| 4.4.251136.1 |
$x^{4} - 32 x^{2} + 109$ |
$4$ |
(4, 0) |
$2^{8}\cdot 3^{2}\cdot 109$ |
$3$ |
$22.386038337$ |
$72.3325652801005$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$3$ |
$69.6184970002$ |
$3$ |
$109$ |
