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 |
4.0.2197.1 |
$x^{4} - x^{3} + 2 x^{2} + 4 x + 3$ |
$4$ |
[0,2] |
$13^{3}$ |
$1$ |
$6.84632504202$ |
$6.846325042023046$ |
✓ |
✓ |
|
$C_4$ (as 4T1) |
trivial |
$2$ |
$1$ |
$2.38952643457$ |
4.4.19773.1 |
$x^{4} - x^{3} - 11 x^{2} - 9 x + 3$ |
$4$ |
[4,0] |
$3^{2}\cdot 13^{3}$ |
$2$ |
$11.8581828179$ |
$11.858182817915045$ |
|
✓ |
✓ |
$C_4$ (as 4T1) |
trivial |
$2$ |
$3$ |
$9.44732277257$ |
4.4.35152.1 |
$x^{4} - 13 x^{2} + 13$ |
$4$ |
[4,0] |
$2^{4}\cdot 13^{3}$ |
$2$ |
$13.692650084$ |
$13.692650084046091$ |
|
✓ |
|
$C_4$ (as 4T1) |
trivial |
$2$ |
$3$ |
$8.93062001005$ |
4.2.50531.1 |
$x^{4} - 2 x^{3} + 8 x^{2} - 7 x - 17$ |
$4$ |
[2,1] |
$-\,13^{3}\cdot 23$ |
$2$ |
$14.9930321835$ |
$32.8338214553794$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$5.45536139347$ |
4.4.63713.1 |
$x^{4} - x^{3} - 11 x^{2} + 4 x + 16$ |
$4$ |
[4,0] |
$13^{3}\cdot 29$ |
$2$ |
$15.8875530497$ |
$36.868588674506164$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$3$ |
$25.8341433054$ |
4.2.94471.1 |
$x^{4} - x^{3} - 11 x^{2} - 22 x - 36$ |
$4$ |
[2,1] |
$-\,13^{3}\cdot 43$ |
$2$ |
$17.5317241267$ |
$44.89435558045543$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$23.3447582997$ |
4.0.105456.1 |
$x^{4} + 13 x^{2} + 39$ |
$4$ |
[0,2] |
$2^{4}\cdot 3\cdot 13^{3}$ |
$3$ |
$18.0205409441$ |
$33.54000593239012$ |
✓ |
|
✓ |
$D_{4}$ (as 4T3) |
$[2, 2]$ |
$2$ |
$1$ |
$2.38952643457$ |
4.4.105456.1 |
$x^{4} - 13 x^{2} + 39$ |
$4$ |
[4,0] |
$2^{4}\cdot 3\cdot 13^{3}$ |
$3$ |
$18.0205409441$ |
$33.54000593239012$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$3$ |
$36.8710665742$ |
4.2.112047.1 |
$x^{4} - 2 x^{3} - 5 x^{2} + 6 x - 43$ |
$4$ |
[2,1] |
$-\,3\cdot 13^{3}\cdot 17$ |
$3$ |
$18.2957433008$ |
$48.892540286148204$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$2$ |
$2$ |
$9.31024114061$ |
4.2.112047.2 |
$x^{4} - x^{3} + 15 x^{2} + 30 x - 36$ |
$4$ |
[2,1] |
$-\,3\cdot 13^{3}\cdot 17$ |
$3$ |
$18.2957433008$ |
$48.892540286148204$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$2$ |
$2$ |
$16.4908377198$ |
4.0.116441.1 |
$x^{4} - 2 x^{3} + 21 x^{2} - 20 x + 48$ |
$4$ |
[0,2] |
$13^{3}\cdot 53$ |
$2$ |
$18.4725342998$ |
$49.84199864366084$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
$[8]$ |
$2$ |
$1$ |
$2.38952643457$ |
4.0.134017.1 |
$x^{4} - x^{3} + 15 x^{2} - 22 x + 68$ |
$4$ |
[0,2] |
$13^{3}\cdot 61$ |
$2$ |
$19.1333040494$ |
$53.471507940612106$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$2$ |
$1$ |
$2.38952643457$ |
4.2.140608.1 |
$x^{4} - 52$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 13^{3}$ |
$2$ |
$19.3643314537$ |
$27.385300168092183$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$24.6378072219$ |
4.2.164775.1 |
$x^{4} - x^{3} + 15 x^{2} - 48 x - 36$ |
$4$ |
[2,1] |
$-\,3\cdot 5^{2}\cdot 13^{3}$ |
$3$ |
$20.1475772711$ |
$26.51570287047805$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$2$ |
$2$ |
$20.6493211639$ |
4.2.202124.1 |
$x^{4} - 2 x^{3} + 8 x^{2} + 6 x - 4$ |
$4$ |
[2,1] |
$-\,2^{2}\cdot 13^{3}\cdot 23$ |
$3$ |
$21.203349455$ |
$52.1204427184053$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$2$ |
$2$ |
$31.0254430133$ |
4.2.208715.1 |
$x^{4} - 13 x + 13$ |
$4$ |
[2,1] |
$-\,5\cdot 13^{3}\cdot 19$ |
$3$ |
$21.3741284772$ |
$66.72972220231422$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2]$ |
$2$ |
$2$ |
$11.7587433585$ |
4.2.226291.1 |
$x^{4} - x^{3} - 11 x^{2} + 43 x - 101$ |
$4$ |
[2,1] |
$-\,13^{3}\cdot 103$ |
$2$ |
$21.8105601957$ |
$69.48261047086733$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$27.9027111237$ |
4.0.237276.1 |
$x^{4} - x^{3} + 15 x^{2} + 17 x + 16$ |
$4$ |
[0,2] |
$2^{2}\cdot 3^{3}\cdot 13^{3}$ |
$3$ |
$22.0705651009$ |
$31.212492494995494$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2, 4]$ |
$2$ |
$1$ |
$9.20933970133$ |
4.0.237276.2 |
$x^{4} + 39$ |
$4$ |
[0,2] |
$2^{2}\cdot 3^{3}\cdot 13^{3}$ |
$3$ |
$22.0705651009$ |
$31.212492494995494$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2, 4]$ |
$2$ |
$1$ |
$6.30951472956$ |
4.4.248261.1 |
$x^{4} - 2 x^{3} - 18 x^{2} + 19 x + 87$ |
$4$ |
[4,0] |
$13^{3}\cdot 113$ |
$2$ |
$22.3216927072$ |
$72.77743347808165$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$3$ |
$38.1400276411$ |
4.2.279019.1 |
$x^{4} - x^{3} - 11 x^{2} - 35 x - 101$ |
$4$ |
[2,1] |
$-\,13^{3}\cdot 127$ |
$2$ |
$22.9830914152$ |
$77.15416486354476$ |
|
|
|
$D_{4}$ (as 4T3) |
$[3]$ |
$2$ |
$2$ |
$9.82816112783$ |
4.0.316368.2 |
$x^{4} + 39 x^{2} + 117$ |
$4$ |
[0,2] |
$2^{4}\cdot 3^{2}\cdot 13^{3}$ |
$3$ |
$23.7163656358$ |
$23.71636563583009$ |
✓ |
✓ |
|
$C_4$ (as 4T1) |
$[2, 2, 2]$ |
$2$ |
$1$ |
$2.38952643457$ |
4.2.322959.1 |
$x^{4} - x^{3} - 24 x^{2} + 69 x - 153$ |
$4$ |
[2,1] |
$-\,3\cdot 7^{2}\cdot 13^{3}$ |
$3$ |
$23.8389349144$ |
$31.37380273734233$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$2$ |
$2$ |
$29.7816556869$ |
4.4.344929.1 |
$x^{4} - x^{3} - 24 x^{2} + 17 x + 107$ |
$4$ |
[4,0] |
$13^{3}\cdot 157$ |
$2$ |
$24.2344086765$ |
$85.78420689860111$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$3$ |
$40.6333360158$ |
4.2.393263.1 |
$x^{4} - x^{3} + 2 x^{2} + 43 x - 153$ |
$4$ |
[2,1] |
$-\,13^{3}\cdot 179$ |
$2$ |
$25.0421015282$ |
$91.5975863110197$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$91.9166714914$ |
4.2.421824.1 |
$x^{4} - 26 x^{2} - 156$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 3\cdot 13^{3}$ |
$3$ |
$25.4848934044$ |
$33.54000593239012$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$2$ |
$2$ |
$13.7583411088$ |
4.2.421824.2 |
$x^{4} + 26 x^{2} - 156$ |
$4$ |
[2,1] |
$-\,2^{6}\cdot 3\cdot 13^{3}$ |
$3$ |
$25.4848934044$ |
$33.54000593239012$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$2$ |
$2$ |
$27.4814481374$ |
4.2.437203.1 |
$x^{4} - x^{3} + 28 x^{2} + 56 x - 153$ |
$4$ |
[2,1] |
$-\,13^{3}\cdot 199$ |
$2$ |
$25.7140677209$ |
$96.57929979879405$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$33.3199608934$ |
4.2.454779.1 |
$x^{4} - 2 x^{3} - 18 x^{2} + 19 x - 173$ |
$4$ |
[2,1] |
$-\,3^{2}\cdot 13^{3}\cdot 23$ |
$3$ |
$25.9686935013$ |
$56.86984696736222$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2, 2]$ |
$2$ |
$2$ |
$8.40482099894$ |
4.2.483340.1 |
$x^{4} - 26 x - 52$ |
$4$ |
[2,1] |
$-\,2^{2}\cdot 5\cdot 11\cdot 13^{3}$ |
$4$ |
$26.367149671$ |
$80.59823339415273$ |
|
|
|
$S_4$ (as 4T5) |
$[2]$ |
$2$ |
$2$ |
$48.0676438985$ |
4.0.494325.1 |
$x^{4} - x^{3} - 11 x^{2} + 30 x + 120$ |
$4$ |
[0,2] |
$3^{2}\cdot 5^{2}\cdot 13^{3}$ |
$3$ |
$26.5157028705$ |
$26.51570287047805$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2, 8]$ |
$2$ |
$1$ |
$8.64159561709$ |
4.0.520689.1 |
$x^{4} - 2 x^{3} + 47 x^{2} - 46 x + 204$ |
$4$ |
[0,2] |
$3\cdot 13^{3}\cdot 79$ |
$3$ |
$26.862386947$ |
$105.39783432170013$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
$[2, 12]$ |
$2$ |
$1$ |
$2.38952643457$ |
4.4.520689.1 |
$x^{4} - x^{3} - 37 x^{2} - 22 x + 172$ |
$4$ |
[4,0] |
$3\cdot 13^{3}\cdot 79$ |
$3$ |
$26.862386947$ |
$105.39783432170013$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$3$ |
$56.7773667534$ |
4.2.551447.1 |
$x^{4} - 13 x^{2} - 13 x + 13$ |
$4$ |
[2,1] |
$-\,13^{3}\cdot 251$ |
$2$ |
$27.2505921972$ |
$108.46618741266329$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2]$ |
$2$ |
$2$ |
$29.9597319626$ |
4.2.551447.2 |
$x^{4} - 2 x^{3} - 31 x^{2} + 32 x - 212$ |
$4$ |
[2,1] |
$-\,13^{3}\cdot 251$ |
$2$ |
$27.2505921972$ |
$108.46618741266329$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$34.9608037836$ |
4.0.562432.1 |
$x^{4} + 13$ |
$4$ |
[0,2] |
$2^{8}\cdot 13^{3}$ |
$2$ |
$27.3853001681$ |
$27.385300168092183$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$2$ |
$1$ |
$10.3107489691$ |
4.0.573417.1 |
$x^{4} - x^{3} + 28 x^{2} - 9 x + 237$ |
$4$ |
[0,2] |
$3^{2}\cdot 13^{3}\cdot 29$ |
$3$ |
$27.51804909$ |
$63.85826878760316$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
$[2, 2, 4]$ |
$2$ |
$1$ |
$2.38952643457$ |
4.4.590993.1 |
$x^{4} - x^{3} - 50 x^{2} + 95 x + 159$ |
$4$ |
[4,0] |
$13^{3}\cdot 269$ |
$2$ |
$27.72653406$ |
$112.28807955565706$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$3$ |
$92.3648049943$ |
4.2.597584.1 |
$x^{4} - 13 x^{2} - 221$ |
$4$ |
[2,1] |
$-\,2^{4}\cdot 13^{3}\cdot 17$ |
$3$ |
$27.803517311$ |
$79.84118395302222$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$2$ |
$2$ |
$11.7300838898$ |
4.2.597584.2 |
$x^{4} + 13 x^{2} - 221$ |
$4$ |
[2,1] |
$-\,2^{4}\cdot 13^{3}\cdot 17$ |
$3$ |
$27.803517311$ |
$79.84118395302222$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$2$ |
$2$ |
$48.7985223314$ |
4.2.621751.1 |
$x^{4} - 2 x^{3} + 21 x^{2} - 20 x - 225$ |
$4$ |
[2,1] |
$-\,13^{3}\cdot 283$ |
$2$ |
$28.0804538705$ |
$115.17301395045637$ |
|
|
|
$D_{4}$ (as 4T3) |
$[5]$ |
$2$ |
$2$ |
$13.8776503115$ |
4.0.634933.1 |
$x^{4} - x^{3} + 54 x^{2} + 56 x + 263$ |
$4$ |
[0,2] |
$13^{3}\cdot 17^{2}$ |
$2$ |
$28.2281212956$ |
$28.228121295572286$ |
✓ |
✓ |
|
$C_4$ (as 4T1) |
$[2, 2]$ |
$2$ |
$1$ |
$2.38952643457$ |
4.2.665691.1 |
$x^{4} - x^{3} + 2 x^{2} + 56 x - 257$ |
$4$ |
[2,1] |
$-\,3\cdot 13^{3}\cdot 101$ |
$3$ |
$28.5639438943$ |
$119.17326241255903$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$2$ |
$2$ |
$16.0119364533$ |
4.2.665691.2 |
$x^{4} - x^{3} - 24 x^{2} + 82 x - 283$ |
$4$ |
[2,1] |
$-\,3\cdot 13^{3}\cdot 101$ |
$3$ |
$28.5639438943$ |
$119.17326241255903$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$2$ |
$2$ |
$27.0933127313$ |
4.2.683267.1 |
$x^{4} - 2 x^{3} + 8 x^{2} - 7 x - 251$ |
$4$ |
[2,1] |
$-\,13^{3}\cdot 311$ |
$2$ |
$28.7506462756$ |
$120.7362572167156$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$2$ |
$18.2521350098$ |
4.4.687661.1 |
$x^{4} - 2 x^{3} - 44 x^{2} + 45 x + 243$ |
$4$ |
[4,0] |
$13^{3}\cdot 313$ |
$2$ |
$28.7967581258$ |
$121.12385454510178$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$3$ |
$117.693823642$ |
4.0.705237.1 |
$x^{4} - 2 x^{3} + 34 x^{2} - 33 x + 269$ |
$4$ |
[0,2] |
$3\cdot 13^{3}\cdot 107$ |
$3$ |
$28.9790251868$ |
$122.661996855225$ |
✓ |
|
✓ |
$D_{4}$ (as 4T3) |
$[2, 4]$ |
$2$ |
$1$ |
$2.38952643457$ |
4.4.705237.1 |
$x^{4} - x^{3} - 37 x^{2} - 9 x + 237$ |
$4$ |
[4,0] |
$3\cdot 13^{3}\cdot 107$ |
$3$ |
$28.9790251868$ |
$122.661996855225$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$2$ |
$3$ |
$38.1259568625$ |
4.0.790920.1 |
$x^{4} - x^{3} + 2 x^{2} - 35 x + 55$ |
$4$ |
[0,2] |
$2^{3}\cdot 3^{2}\cdot 5\cdot 13^{3}$ |
$4$ |
$29.8217509811$ |
$74.99773323057053$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[2, 8]$ |
$2$ |
$1$ |
$3.10714061537$ |
4.0.790920.4 |
$x^{4} - x^{3} + 15 x^{2} + 95 x + 250$ |
$4$ |
[0,2] |
$2^{3}\cdot 3^{2}\cdot 5\cdot 13^{3}$ |
$4$ |
$29.8217509811$ |
$74.99773323057053$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2, 8]$ |
$2$ |
$1$ |
$9.50012732134$ |