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 |
$4$ |
$1$ |
$0.732857675974$ |
4.0.320.1 |
$x^{4} - 2 x^{3} + 2$ |
$4$ |
[0,2] |
$2^{6}\cdot 5$ |
$2$ |
$4.22948505376$ |
$6.324555320336759$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$4$ |
$1$ |
$1.06127506191$ |
4.0.333.1 |
$x^{4} - x^{3} - 2 x^{2} + 3$ |
$4$ |
[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 |
$2$ |
$1$ |
$0.632974319201$ |
4.0.400.1 |
$x^{4} + 3 x^{2} + 1$ |
$4$ |
[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 |
$6$ |
$1$ |
$1.56679923697$ |
4.0.512.1 |
$x^{4} - 2 x^{2} + 2$ |
$4$ |
[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 |
$6$ |
$1$ |
$2.11437619251$ |
4.0.576.1 |
$x^{4} - 2 x^{2} + 4$ |
$4$ |
[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 |
$2$ |
$1$ |
$0.767197218251$ |
4.0.656.1 |
$x^{4} - 2 x^{3} - x^{2} + 2 x + 2$ |
$4$ |
[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 |
$6$ |
$1$ |
$2.0335456547$ |
4.0.697.1 |
$x^{4} - x^{3} + 2 x^{2} - x + 2$ |
$4$ |
[0,2] |
$17\cdot 41$ |
$2$ |
$5.13816675137$ |
$26.40075756488817$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$1.03324185123$ |
4.0.761.1 |
$x^{4} - x^{3} + x^{2} + 2 x + 1$ |
$4$ |
[0,2] |
$761$ |
$1$ |
$5.25225936605$ |
$27.586228448267445$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$0.990699508768$ |
4.0.784.1 |
$x^{4} - 3 x^{2} + 4$ |
$4$ |
[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 |
$6$ |
$1$ |
$2.41883810412$ |
4.0.873.1 |
$x^{4} - x^{3} - 4 x^{2} + x + 7$ |
$4$ |
[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 |
$2$ |
$1$ |
$1.29274426912$ |
4.0.981.1 |
$x^{4} - x^{3} - 4 x^{2} + 4 x + 7$ |
$4$ |
[0,2] |
$3^{2}\cdot 109$ |
$2$ |
$5.59650958426$ |
$18.083141320025124$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$2.15221875719$ |
4.0.985.1 |
$x^{4} - x^{3} + 2 x^{2} - 3 x + 2$ |
$4$ |
[0,2] |
$5\cdot 197$ |
$2$ |
$5.6022057846$ |
$31.38470965295043$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$1.34361173004$ |
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 |
$6$ |
$1$ |
$3.39588318368$ |
4.0.1008.2 |
$x^{4} + 5 x^{2} + 7$ |
$4$ |
[0,2] |
$2^{4}\cdot 3^{2}\cdot 7$ |
$3$ |
$5.63462649452$ |
$12.96148139681572$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$2.41611328691$ |
4.0.1016.1 |
$x^{4} - x^{3} + x^{2} - 2 x + 2$ |
$4$ |
[0,2] |
$2^{3}\cdot 127$ |
$2$ |
$5.64577318895$ |
$31.874754901018456$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$1.27428442991$ |
4.0.1025.1 |
$x^{4} - x^{3} + 3 x^{2} - 2 x + 4$ |
$4$ |
[0,2] |
$5^{2}\cdot 41$ |
$2$ |
$5.65823481195$ |
$14.317821063276353$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$1$ |
$0.962423650119$ |
4.0.1040.1 |
$x^{4} - 3 x^{2} - 2 x + 5$ |
$4$ |
[0,2] |
$2^{4}\cdot 5\cdot 13$ |
$3$ |
$5.67882302887$ |
$16.1245154965971$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$4$ |
$1$ |
$1.8522660627$ |
4.0.1040.2 |
$x^{4} + x^{2} - 4 x + 4$ |
$4$ |
[0,2] |
$2^{4}\cdot 5\cdot 13$ |
$3$ |
$5.67882302887$ |
$16.1245154965971$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$4$ |
$1$ |
$3.46864904298$ |
4.0.1076.1 |
$x^{4} - x^{3} + 3 x^{2} - 3 x + 2$ |
$4$ |
[0,2] |
$2^{2}\cdot 269$ |
$2$ |
$5.7273413495$ |
$26.035313035249676$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$1.77702641037$ |
4.0.1088.1 |
$x^{4} - 2 x^{3} + x^{2} - 2 x + 3$ |
$4$ |
[0,2] |
$2^{6}\cdot 17$ |
$2$ |
$5.74324342205$ |
$11.661903789690601$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$1$ |
$1.17996602392$ |
4.0.1088.2 |
$x^{4} - 2 x^{3} + 5 x^{2} - 4 x + 2$ |
$4$ |
[0,2] |
$2^{6}\cdot 17$ |
$2$ |
$5.74324342205$ |
$11.661903789690601$ |
✓ |
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$1$ |
$1.76274717404$ |
4.0.1089.1 |
$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$ |
$4$ |
[0,2] |
$3^{2}\cdot 11^{2}$ |
$2$ |
$5.74456264654$ |
$5.744562646538029$ |
✓ |
✓ |
|
$C_2^2$ (as 4T2) |
trivial |
$6$ |
$1$ |
$3.82816847133$ |
4.0.1129.1 |
$x^{4} - x^{3} - x + 2$ |
$4$ |
[0,2] |
$1129$ |
$1$ |
$5.79660204196$ |
$33.60059523282288$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$2$ |
$1$ |
$1.26901928105$ |
4.0.1161.1 |
$x^{4} - x^{3} + 6 x^{2} - x + 7$ |
$4$ |
[0,2] |
$3^{3}\cdot 43$ |
$2$ |
$5.83724678892$ |
$14.947727391693533$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$3.48142350173$ |
4.0.1168.1 |
$x^{4} - x^{2} - 4 x + 5$ |
$4$ |
[0,2] |
$2^{4}\cdot 73$ |
$2$ |
$5.84602557138$ |
$17.08800749063506$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$4$ |
$1$ |
$1.8902413596$ |
4.0.1197.1 |
$x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 7$ |
$4$ |
[0,2] |
$3^{2}\cdot 7\cdot 19$ |
$3$ |
$5.8819799204$ |
$19.974984355438178$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$2.3192389338$ |
4.0.1197.2 |
$x^{4} - x^{3} - 5 x^{2} + 3 x + 9$ |
$4$ |
[0,2] |
$3^{2}\cdot 7\cdot 19$ |
$3$ |
$5.8819799204$ |
$19.974984355438178$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$6$ |
$1$ |
$3.57713438377$ |
4.0.1225.1 |
$x^{4} - x^{3} + 5 x^{2} + 2 x + 4$ |
$4$ |
[0,2] |
$5^{2}\cdot 7^{2}$ |
$2$ |
$5.9160797831$ |
$5.916079783099616$ |
✓ |
✓ |
|
$C_2^2$ (as 4T2) |
trivial |
$2$ |
$1$ |
$0.962423650119$ |