| 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 $p$ |
| 4.2.1323.1 |
$x^{4} - x^{3} - 3 x^{2} - x + 1$ |
$4$ |
(2, 1) |
$-\,3^{3}\cdot 7^{2}$ |
$2$ |
$6.03100878452$ |
$6.03100878451909$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$1.35144993247$ |
$1$ |
$7$ |
| 4.2.1728.1 |
$x^{4} - 2 x^{3} - 2 x + 1$ |
$4$ |
(2, 1) |
$-\,2^{6}\cdot 3^{3}$ |
$2$ |
$6.44741959094$ |
$6.4474195909412515$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$2.18995070591$ |
$1$ |
$3$ |
| 4.0.2889.1 |
$x^{4} - x^{3} + 3 x^{2} + 2 x + 1$ |
$4$ |
(0, 2) |
$3^{3}\cdot 107$ |
$2$ |
$7.33139949815$ |
$23.579404344249447$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$1.33243180303$ |
$0$ |
$107$ |
| 4.2.2943.1 |
$x^{4} - x^{3} - x - 2$ |
$4$ |
(2, 1) |
$-\,3^{3}\cdot 109$ |
$2$ |
$7.36542082709$ |
$23.7987523638325$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$3.1893652555$ |
$1$ |
$109$ |
| 4.4.4752.1 |
$x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1$ |
$4$ |
(4, 0) |
$2^{4}\cdot 3^{3}\cdot 11$ |
$3$ |
$8.30269545139$ |
$15.120539229772588$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$3$ |
$3.7053371595$ |
$3$ |
$11$ |
| 4.0.4752.1 |
$x^{4} + 3 x^{2} - 6 x + 6$ |
$4$ |
(0, 2) |
$2^{4}\cdot 3^{3}\cdot 11$ |
$3$ |
$8.30269545139$ |
$15.120539229772588$ |
✓ |
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$2.63391579385$ |
$0$ |
$11$ |
| 4.0.5076.1 |
$x^{4} - x^{3} + 3 x^{2} - x + 4$ |
$4$ |
(0, 2) |
$2^{2}\cdot 3^{3}\cdot 47$ |
$3$ |
$8.44073767643$ |
$24.807130641975725$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$2.98491079846$ |
$0$ |
$47$ |
| 4.0.5400.1 |
$x^{4} - x^{3} + 2 x + 4$ |
$4$ |
(0, 2) |
$2^{3}\cdot 3^{3}\cdot 5^{2}$ |
$3$ |
$8.5723212891$ |
$14.416868484808525$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$4.34612583115$ |
$0$ |
$5$ |
| 4.0.5400.2 |
$x^{4} - 3 x^{2} + 6$ |
$4$ |
(0, 2) |
$2^{3}\cdot 3^{3}\cdot 5^{2}$ |
$3$ |
$8.5723212891$ |
$14.416868484808525$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$2.28206666727$ |
$0$ |
$5$ |
| 4.2.5616.1 |
$x^{4} - 2 x^{3} + 3 x^{2} - 2 x - 2$ |
$4$ |
(2, 1) |
$-\,2^{4}\cdot 3^{3}\cdot 13$ |
$3$ |
$8.65678785583$ |
$16.437759153264928$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$7.10823929493$ |
$1$ |
$13$ |
| 4.2.5616.2 |
$x^{4} - 3 x^{2} - 6 x - 3$ |
$4$ |
(2, 1) |
$-\,2^{4}\cdot 3^{3}\cdot 13$ |
$3$ |
$8.65678785583$ |
$16.437759153264928$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$3.25525922497$ |
$1$ |
$13$ |
| 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 |
$[2]$ |
$2$ |
$2$ |
$5.24584468864$ |
$1$ |
$3$ |
| 4.2.7803.1 |
$x^{4} - x^{3} - 4 x + 1$ |
$4$ |
(2, 1) |
$-\,3^{3}\cdot 17^{2}$ |
$2$ |
$9.39864837017$ |
$15.070935874553902$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$3.39817572226$ |
$1$ |
$17$ |
| 4.2.8856.1 |
$x^{4} - x^{3} - 3 x^{2} + 2 x - 2$ |
$4$ |
(2, 1) |
$-\,2^{3}\cdot 3^{3}\cdot 41$ |
$3$ |
$9.70084134744$ |
$41.28362865165531$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$7.28883660653$ |
$1$ |
$41$ |
| 4.2.9099.1 |
$x^{4} - 3 x - 3$ |
$4$ |
(2, 1) |
$-\,3^{3}\cdot 337$ |
$2$ |
$9.76671278238$ |
$41.846187000157315$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$4.01249103902$ |
$1$ |
$337$ |
| 4.2.9747.1 |
$x^{4} + 3 x^{2} - 12$ |
$4$ |
(2, 1) |
$-\,3^{3}\cdot 19^{2}$ |
$2$ |
$9.93614090235$ |
$9.93614090235369$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$11.2208440566$ |
$1$ |
$19$ |
| 4.2.10476.1 |
$x^{4} - 6 x - 3$ |
$4$ |
(2, 1) |
$-\,2^{2}\cdot 3^{3}\cdot 97$ |
$3$ |
$10.1169329668$ |
$35.63801218181345$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$11.2663192331$ |
$1$ |
$97$ |
| 4.2.10719.1 |
$x^{4} - 3 x^{2} - 3 x - 3$ |
$4$ |
(2, 1) |
$-\,3^{3}\cdot 397$ |
$2$ |
$10.1750972261$ |
$45.41885634639574$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$7.96194872745$ |
$1$ |
$397$ |
| 4.2.10800.2 |
$x^{4} - 2 x^{3} - 8 x + 4$ |
$4$ |
(2, 1) |
$-\,2^{4}\cdot 3^{3}\cdot 5^{2}$ |
$3$ |
$10.1942654691$ |
$14.963135389464592$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$8.16300392804$ |
$1$ |
$5$ |
| 4.2.11448.1 |
$x^{4} - x^{3} - 3 x^{2} + 8 x - 8$ |
$4$ |
(2, 1) |
$-\,2^{3}\cdot 3^{3}\cdot 53$ |
$3$ |
$10.3438545524$ |
$46.93792312435236$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$8.19666709785$ |
$1$ |
$53$ |
| 4.0.13500.1 |
$x^{4} - x^{3} + 6 x^{2} - x + 1$ |
$4$ |
(0, 2) |
$2^{2}\cdot 3^{3}\cdot 5^{3}$ |
$3$ |
$10.7791233589$ |
$15.243982444638442$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$1.7400216453$ |
$0$ |
$5$ |
| 4.0.13500.2 |
$x^{4} + 15$ |
$4$ |
(0, 2) |
$2^{2}\cdot 3^{3}\cdot 5^{3}$ |
$3$ |
$10.7791233589$ |
$15.243982444638442$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$4.14262643348$ |
$0$ |
$5$ |
| 4.4.13824.1 |
$x^{4} - 6 x^{2} + 6$ |
$4$ |
(4, 0) |
$2^{9}\cdot 3^{3}$ |
$2$ |
$10.8432240433$ |
$15.33463450191054$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$3$ |
$10.1286120835$ |
$3$ |
$3$ |
| 4.0.13824.1 |
$x^{4} + 6 x^{2} + 6$ |
$4$ |
(0, 2) |
$2^{9}\cdot 3^{3}$ |
$2$ |
$10.8432240433$ |
$15.33463450191054$ |
✓ |
|
✓ |
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$2.63391579385$ |
$0$ |
$3$ |
| 4.2.15336.1 |
$x^{4} - x^{3} + 5 x + 1$ |
$4$ |
(2, 1) |
$-\,2^{3}\cdot 3^{3}\cdot 71$ |
$3$ |
$11.1282795006$ |
$54.326923123782436$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$10.6359393369$ |
$1$ |
$71$ |
| 4.0.15525.1 |
$x^{4} - 2 x^{3} + x + 4$ |
$4$ |
(0, 2) |
$3^{3}\cdot 5^{2}\cdot 23$ |
$3$ |
$11.1624082502$ |
$24.444989846822654$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$5.80853122949$ |
$0$ |
$23$ |
| 4.0.15525.2 |
$x^{4} - x^{3} + 9 x^{2} + 5 x + 10$ |
$4$ |
(0, 2) |
$3^{3}\cdot 5^{2}\cdot 23$ |
$3$ |
$11.1624082502$ |
$24.444989846822654$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$6.96755413432$ |
$0$ |
$23$ |
| 4.2.15984.2 |
$x^{4} - 2 x^{3} - 3 x^{2} + 4 x - 8$ |
$4$ |
(2, 1) |
$-\,2^{4}\cdot 3^{3}\cdot 37$ |
$3$ |
$11.2440137422$ |
$27.731400227189784$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$13.888497916$ |
$1$ |
$37$ |
| 4.2.15984.3 |
$x^{4} + 3 x^{2} - 12 x - 3$ |
$4$ |
(2, 1) |
$-\,2^{4}\cdot 3^{3}\cdot 37$ |
$3$ |
$11.2440137422$ |
$27.731400227189784$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$2$ |
$5.58804654399$ |
$1$ |
$37$ |
| 4.2.16308.1 |
$x^{4} - x^{3} - 3 x^{2} - 4 x + 4$ |
$4$ |
(2, 1) |
$-\,2^{2}\cdot 3^{3}\cdot 151$ |
$3$ |
$11.3005654692$ |
$56.02210334604374$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$10.0975970712$ |
$1$ |
$151$ |
| 4.2.18252.1 |
$x^{4} - x^{3} - 6 x^{2} - 4 x - 2$ |
$4$ |
(2, 1) |
$-\,2^{2}\cdot 3^{3}\cdot 13^{2}$ |
$3$ |
$11.6232509648$ |
$20.005826881210375$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$15.3739530492$ |
$1$ |
$13$ |
| 4.2.18576.1 |
$x^{4} - 2 x^{3} + 3 x^{2} + 10 x + 1$ |
$4$ |
(2, 1) |
$-\,2^{4}\cdot 3^{3}\cdot 43$ |
$3$ |
$11.6744935778$ |
$42.27855760797761$ |
|
|
|
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$7.20576533044$ |
$1$ |
$43$ |
| 4.0.19008.1 |
$x^{4} - 2 x^{3} + 3 x^{2} + 4 x + 4$ |
$4$ |
(0, 2) |
$2^{6}\cdot 3^{3}\cdot 11$ |
$3$ |
$11.7417845116$ |
$21.383671649138826$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$7.61136876171$ |
$0$ |
$11$ |
| 4.0.19008.2 |
$x^{4} - 12 x + 15$ |
$4$ |
(0, 2) |
$2^{6}\cdot 3^{3}\cdot 11$ |
$3$ |
$11.7417845116$ |
$21.383671649138826$ |
|
|
|
$D_{4}$ (as 4T3) |
$[2]$ |
$[2]$ |
$2$ |
$1$ |
$3.2893017342$ |
$0$ |
$11$ |
| 4.2.19791.1 |
$x^{4} - 2 x^{3} - 3 x^{2} + x - 2$ |
$4$ |
(2, 1) |
$-\,3^{3}\cdot 733$ |
$2$ |
$11.8608806188$ |
$61.715311923735435$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$10.6749623213$ |
$1$ |
$733$ |
| 4.0.20304.1 |
$x^{4} - 6 x + 6$ |
$4$ |
(0, 2) |
$2^{4}\cdot 3^{3}\cdot 47$ |
$3$ |
$11.9370056984$ |
$35.082580597443254$ |
|
|
✓ |
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$1.99997356006$ |
$0$ |
$47$ |
| 4.0.20628.1 |
$x^{4} - x^{3} + 3 x^{2} + 5 x + 16$ |
$4$ |
(0, 2) |
$2^{2}\cdot 3^{3}\cdot 191$ |
$3$ |
$11.9843443894$ |
$50.00855300493292$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$3.30120913265$ |
$0$ |
$191$ |
| 4.0.20628.2 |
$x^{4} - x^{3} + 8 x + 16$ |
$4$ |
(0, 2) |
$2^{2}\cdot 3^{3}\cdot 191$ |
$3$ |
$11.9843443894$ |
$50.00855300493292$ |
|
|
|
$S_4$ (as 4T5) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$3.64246303658$ |
$0$ |
$191$ |
| 4.2.20844.1 |
$x^{4} - x^{3} - 6 x^{2} + 8 x + 4$ |
$4$ |
(2, 1) |
$-\,2^{2}\cdot 3^{3}\cdot 193$ |
$3$ |
$12.0155945748$ |
$50.26969604951383$ |
|
|
|
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$16.6122381609$ |
$1$ |
$193$ |
| 4.2.21168.1 |
$x^{4} + 3 x^{2} - 3$ |
$4$ |
(2, 1) |
$-\,2^{4}\cdot 3^{3}\cdot 7^{2}$ |
$3$ |
$12.062017569$ |
$12.06201756903818$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[2]$ |
$[2, 2]$ |
$2$ |
$2$ |
$4.44887926078$ |
$1$ |
$7$ |
| 4.2.21168.2 |
$x^{4} - 6 x^{2} - 6 x - 3$ |
$4$ |
(2, 1) |
$-\,2^{4}\cdot 3^{3}\cdot 7^{2}$ |
$3$ |
$12.062017569$ |
$18.725839469079336$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$10.4252997029$ |
$1$ |
$7$ |
| 4.0.21600.1 |
$x^{4} + 3 x^{2} + 6$ |
$4$ |
(0, 2) |
$2^{5}\cdot 3^{3}\cdot 5^{2}$ |
$3$ |
$12.1230930281$ |
$20.38853093816547$ |
|
|
✓ |
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$3.67808759807$ |
$0$ |
$5$ |
| 4.0.21600.2 |
$x^{4} + 9 x^{2} + 24$ |
$4$ |
(0, 2) |
$2^{5}\cdot 3^{3}\cdot 5^{2}$ |
$3$ |
$12.1230930281$ |
$20.38853093816547$ |
|
|
|
$D_{4}$ (as 4T3) |
$[4]$ |
$[4]$ |
$2$ |
$1$ |
$2.87591184041$ |
$0$ |
$5$ |
| 4.4.23301.1 |
$x^{4} - x^{3} - 6 x^{2} + 2 x + 1$ |
$4$ |
(4, 0) |
$3^{3}\cdot 863$ |
$2$ |
$12.3550253257$ |
$66.96476342671437$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$3$ |
$9.84593152366$ |
$3$ |
$863$ |
| 4.2.23355.1 |
$x^{4} - x^{3} + 2 x - 5$ |
$4$ |
(2, 1) |
$-\,3^{3}\cdot 5\cdot 173$ |
$3$ |
$12.362177297$ |
$67.0423138446253$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$8.86333129188$ |
$1$ |
$173$ |
| 4.2.23436.1 |
$x^{4} - x^{3} - 3 x^{2} + 5 x + 4$ |
$4$ |
(2, 1) |
$-\,2^{2}\cdot 3^{3}\cdot 7\cdot 31$ |
$4$ |
$12.3728820356$ |
$53.303714205206774$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$19.2860009917$ |
$1$ |
$31$ |
| 4.2.23679.1 |
$x^{4} - 3 x^{2} - 12 x - 3$ |
$4$ |
(2, 1) |
$-\,3^{3}\cdot 877$ |
$2$ |
$12.4048306413$ |
$67.5057454940964$ |
|
|
|
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$9.59840223489$ |
$1$ |
$877$ |
| 4.2.24084.1 |
$x^{4} - x^{3} + 3 x^{2} - 4 x - 2$ |
$4$ |
(2, 1) |
$-\,2^{2}\cdot 3^{3}\cdot 223$ |
$3$ |
$12.4575361078$ |
$68.08059900628311$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$14.3553980542$ |
$1$ |
$223$ |
| 4.2.25056.1 |
$x^{4} - 2 x^{3} - 2 x - 17$ |
$4$ |
(2, 1) |
$-\,2^{5}\cdot 3^{3}\cdot 29$ |
$3$ |
$12.5813700168$ |
$49.10208472291046$ |
|
|
✓ |
$S_4$ (as 4T5) |
trivial |
$[2]$ |
$2$ |
$2$ |
$19.9832224169$ |
$1$ |
$29$ |
| 4.4.25488.1 |
$x^{4} - 9 x^{2} - 6 x + 6$ |
$4$ |
(4, 0) |
$2^{4}\cdot 3^{3}\cdot 59$ |
$3$ |
$12.6352529198$ |
$35.01845187552935$ |
|
|
|
$D_{4}$ (as 4T3) |
trivial |
$[2]$ |
$2$ |
$3$ |
$23.0109465771$ |
$3$ |
$59$ |
