| 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 |
| 4.0.117.1 |
$x^{4} - x^{3} - x^{2} + x + 1$ |
$4$ |
[0,2] |
$3^{2}\cdot 13$ |
$2$ |
$3.28886816799$ |
$6.244997998398398$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$0.543535072498$ |
| 4.0.125.1 |
$x^{4} - x^{3} + x^{2} - x + 1$ |
$4$ |
[0,2] |
$5^{3}$ |
$1$ |
$3.34370152488$ |
$3.34370152488211$ |
✓ |
✓ |
✓ |
$C_4$ (as 4T1) |
trivial |
$10$ |
$1$ |
$0.962423650119$ |
| 4.0.144.1 |
$x^{4} - x^{2} + 1$ |
$4$ |
[0,2] |
$2^{4}\cdot 3^{2}$ |
$2$ |
$3.46410161514$ |
$3.4641016151377544$ |
✓ |
✓ |
✓ |
$C_2^2$ (as 4T2) |
trivial |
$12$ |
$1$ |
$1.31695789692$ |
| 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 |
$6$ |
$1$ |
$0.862554627662$ |
| 4.0.225.1 |
$x^{4} - x^{3} + 2 x^{2} + x + 1$ |
$4$ |
[0,2] |
$3^{2}\cdot 5^{2}$ |
$2$ |
$3.87298334621$ |
$3.872983346207417$ |
✓ |
✓ |
|
$C_2^2$ (as 4T2) |
trivial |
$6$ |
$1$ |
$0.962423650119$ |
| 4.0.229.1 |
$x^{4} - x + 1$ |
$4$ |
[0,2] |
$229$ |
$1$ |
$3.89008302616$ |
$15.132745950421556$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$0.337377803572$ |
| 4.0.256.1 |
$x^{4} + 1$ |
$4$ |
[0,2] |
$2^{8}$ |
$1$ |
$4.0$ |
$4.0$ |
✓ |
✓ |
✓ |
$C_2^2$ (as 4T2) |
trivial |
$8$ |
$1$ |
$1.76274717404$ |
| 4.0.257.1 |
$x^{4} + x^{2} - x + 1$ |
$4$ |
[0,2] |
$257$ |
$1$ |
$4.00390054096$ |
$16.0312195418814$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$0.442137828732$ |
| 4.0.272.1 |
$x^{4} + x^{2} - 2 x + 1$ |
$4$ |
[0,2] |
$2^{4}\cdot 17$ |
$2$ |
$4.06108636974$ |
$8.246211251235321$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$4$ |
$1$ |
$0.732857675974$ |
| 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 |
$2$ |
$2$ |
$0.37819933246$ |
| 4.0.320.1 |
$x^{4} - 2 x^{3} + 2$ |
$4$ |
[0,2] |
$2^{6}\cdot 5$ |
$2$ |
$4.22948505376$ |
$6.324555320336759$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$4$ |
$1$ |
$1.06127506191$ |
| 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 |
$2$ |
$2$ |
$0.432203091185$ |
| 4.0.333.1 |
$x^{4} - x^{3} - 2 x^{2} + 3$ |
$4$ |
[0,2] |
$3^{2}\cdot 37$ |
$2$ |
$4.27180144563$ |
$10.535653752852738$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$1.46078161311$ |
| 4.0.392.1 |
$x^{4} - x^{3} + x + 1$ |
$4$ |
[0,2] |
$2^{3}\cdot 7^{2}$ |
$2$ |
$4.44960558625$ |
$7.483314773547883$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$1$ |
$0.632974319201$ |
| 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 |
$2$ |
$2$ |
$0.51069810943$ |
| 4.0.400.1 |
$x^{4} + 3 x^{2} + 1$ |
$4$ |
[0,2] |
$2^{4}\cdot 5^{2}$ |
$2$ |
$4.472135955$ |
$4.47213595499958$ |
✓ |
✓ |
✓ |
$C_2^2$ (as 4T2) |
trivial |
$4$ |
$1$ |
$0.962423650119$ |
| 4.0.432.1 |
$x^{4} - 3 x^{2} + 3$ |
$4$ |
[0,2] |
$2^{4}\cdot 3^{3}$ |
$2$ |
$4.55901411391$ |
$6.4474195909412515$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$1.66288589106$ |
| 4.0.441.1 |
$x^{4} - x^{3} - x^{2} - 2 x + 4$ |
$4$ |
[0,2] |
$3^{2}\cdot 7^{2}$ |
$2$ |
$4.58257569496$ |
$4.58257569495584$ |
✓ |
✓ |
|
$C_2^2$ (as 4T2) |
trivial |
$6$ |
$1$ |
$1.56679923697$ |
| 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 |
$2$ |
$2$ |
$0.649395711926$ |
| 4.0.512.1 |
$x^{4} - 2 x^{2} + 2$ |
$4$ |
[0,2] |
$2^{9}$ |
$1$ |
$4.75682846001$ |
$6.727171322029716$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$4$ |
$1$ |
$1.52857091948$ |
| 4.0.513.1 |
$x^{4} - x^{3} - 3 x^{2} + 2 x + 4$ |
$4$ |
[0,2] |
$3^{3}\cdot 19$ |
$2$ |
$4.75914943092$ |
$9.93614090235369$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$1.96497837511$ |
| 4.0.549.1 |
$x^{4} - 2 x^{3} - 2 x^{2} + 3 x + 3$ |
$4$ |
[0,2] |
$3^{2}\cdot 61$ |
$2$ |
$4.84053189512$ |
$13.527749258468683$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$2.11437619251$ |
| 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 |
$2$ |
$2$ |
$0.704267195983$ |
| 4.0.576.1 |
$x^{4} - 2 x^{2} + 4$ |
$4$ |
[0,2] |
$2^{6}\cdot 3^{2}$ |
$2$ |
$4.89897948557$ |
$4.898979485566356$ |
✓ |
✓ |
|
$C_2^2$ (as 4T2) |
trivial |
$6$ |
$1$ |
$2.29243166956$ |
| 4.0.576.2 |
$x^{4} + 2 x^{2} + 4$ |
$4$ |
[0,2] |
$2^{6}\cdot 3^{2}$ |
$2$ |
$4.89897948557$ |
$4.898979485566356$ |
✓ |
✓ |
|
$C_2^2$ (as 4T2) |
trivial |
$6$ |
$1$ |
$1.76274717404$ |
| 4.0.592.1 |
$x^{4} + 2 x^{2} - 2 x + 1$ |
$4$ |
[0,2] |
$2^{4}\cdot 37$ |
$2$ |
$4.93265142912$ |
$13.655340178276088$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$0.917818607957$ |
| 4.0.605.1 |
$x^{4} - x^{3} + x^{2} + x + 1$ |
$4$ |
[0,2] |
$5\cdot 11^{2}$ |
$2$ |
$4.95951083803$ |
$7.416198487095663$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$1$ |
$0.767197218251$ |
| 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 |
$2$ |
$2$ |
$0.720184467868$ |
| 4.0.656.1 |
$x^{4} - 2 x^{3} - x^{2} + 2 x + 2$ |
$4$ |
[0,2] |
$2^{4}\cdot 41$ |
$2$ |
$5.06087906887$ |
$12.806248474865697$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$4$ |
$1$ |
$2.42610665212$ |
| 4.0.657.1 |
$x^{4} - x^{3} + 5 x^{2} - 2 x + 4$ |
$4$ |
[0,2] |
$3^{2}\cdot 73$ |
$2$ |
$5.06280665599$ |
$14.798648586948742$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$2.0335456547$ |
| 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 |
$2$ |
$2$ |
$0.998743211853$ |
| 4.0.697.1 |
$x^{4} - x^{3} + 2 x^{2} - x + 2$ |
$4$ |
[0,2] |
$17\cdot 41$ |
$2$ |
$5.13816675137$ |
$26.40075756488817$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$1.03324185123$ |
| 4.4.725.1 |
$x^{4} - x^{3} - 3 x^{2} + x + 1$ |
$4$ |
[4,0] |
$5^{2}\cdot 29$ |
$2$ |
$5.18900992827$ |
$12.041594578792296$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$3$ |
$0.825068847935$ |
| 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 |
$2$ |
$2$ |
$1.06794059626$ |
| 4.0.761.1 |
$x^{4} - x^{3} + x^{2} + 2 x + 1$ |
$4$ |
[0,2] |
$761$ |
$1$ |
$5.25225936605$ |
$27.586228448267445$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$0.990699508768$ |
| 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 |
$2$ |
$2$ |
$0.867905309463$ |
| 4.0.784.1 |
$x^{4} - 3 x^{2} + 4$ |
$4$ |
[0,2] |
$2^{4}\cdot 7^{2}$ |
$2$ |
$5.29150262213$ |
$5.291502622129181$ |
✓ |
✓ |
|
$C_2^2$ (as 4T2) |
trivial |
$4$ |
$1$ |
$2.76865938331$ |
| 4.0.788.1 |
$x^{4} - x^{3} + 2 x^{2} - 2 x + 2$ |
$4$ |
[0,2] |
$2^{2}\cdot 197$ |
$2$ |
$5.29823911269$ |
$22.280235493786417$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$1.39411358598$ |
| 4.0.832.1 |
$x^{4} - 2 x^{3} - 2 x + 5$ |
$4$ |
[0,2] |
$2^{6}\cdot 13$ |
$2$ |
$5.37069922857$ |
$10.198039027185569$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$4$ |
$1$ |
$2.13862208632$ |
| 4.0.837.1 |
$x^{4} - x^{3} - 3 x^{2} - x + 7$ |
$4$ |
[0,2] |
$3^{3}\cdot 31$ |
$2$ |
$5.37875006697$ |
$12.691758156532355$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$2.41883810412$ |
| 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 |
$2$ |
$2$ |
$0.992994938315$ |
| 4.0.873.1 |
$x^{4} - x^{3} - 4 x^{2} + x + 7$ |
$4$ |
[0,2] |
$3^{2}\cdot 97$ |
$2$ |
$5.43567598422$ |
$17.05872210923198$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$2.42800256046$ |
| 4.0.892.1 |
$x^{4} - x^{3} - x^{2} + 2$ |
$4$ |
[0,2] |
$2^{2}\cdot 223$ |
$2$ |
$5.46501317895$ |
$29.866369046136157$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$1.29274426912$ |
| 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 |
$2$ |
$2$ |
$0.990699215006$ |
| 4.0.981.1 |
$x^{4} - x^{3} - 4 x^{2} + 4 x + 7$ |
$4$ |
[0,2] |
$3^{2}\cdot 109$ |
$2$ |
$5.59650958426$ |
$18.083141320025124$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$2.15221875719$ |
