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 |
19.1.756...064.1 |
$x^{19} - 2 x^{18} + 8 x^{17} - 12 x^{16} + 43 x^{15} - 74 x^{14} + 108 x^{13} - 18 x^{12} - 25 x^{11} + 338 x^{9} - 70 x^{8} - 264 x^{7} - 128 x^{6} + 120 x^{5} + 98 x^{4} + 43 x^{3} - 22 x^{2} - 31 x - 8$ |
$19$ |
[1,9] |
$-\,2^{18}\cdot 19^{16}$ |
$2$ |
$23.0160848178$ |
$27.39680150407798$ |
|
|
? |
$F_{19}$ (as 19T6) |
trivial |
$2$ |
$9$ |
$540383.098056$ |
19.1.518...976.1 |
$x^{19} - 2$ |
$19$ |
[1,9] |
$-\,2^{18}\cdot 19^{19}$ |
$2$ |
$36.6386879218$ |
$42.41345857356653$ |
|
|
✓ |
$F_{19}$ (as 19T6) |
trivial |
$2$ |
$9$ |
$47980973.6594$ |
19.1.386...375.1 |
$x^{19} + 19 x^{17} + 152 x^{15} + 665 x^{13} + 1729 x^{11} + 2717 x^{9} + 2508 x^{7} + 1254 x^{5} + 285 x^{3} + 19 x - 1$ |
$19$ |
[1,9] |
$-\,5^{9}\cdot 19^{19}$ |
$2$ |
$40.7234586539$ |
$49.18156889081904$ |
|
|
✓ |
$F_{19}$ (as 19T6) |
trivial |
$2$ |
$9$ |
$123041861.807$ |
19.1.766...731.1 |
$x^{19} - 3$ |
$19$ |
[1,9] |
$-\,3^{18}\cdot 19^{19}$ |
$2$ |
$53.7976382998$ |
$62.27689998761435$ |
|
|
✓ |
$F_{19}$ (as 19T6) |
trivial |
$2$ |
$9$ |
$1832073247.14$ |
19.1.754...875.1 |
$x^{19} - 5$ |
$19$ |
[1,9] |
$-\,5^{18}\cdot 19^{19}$ |
$2$ |
$87.2842149862$ |
$101.04143042305182$ |
|
|
✓ |
$F_{19}$ (as 19T6) |
trivial |
$2$ |
$9$ |
$183047330087$ |
19.1.200...264.1 |
$x^{19} - 6$ |
$19$ |
[1,9] |
$-\,2^{18}\cdot 3^{18}\cdot 19^{19}$ |
$3$ |
$103.740783189$ |
$120.09178438864606$ |
|
|
✓ |
$F_{19}$ (as 19T6) |
trivial |
$2$ |
$9$ |
$929704933317$ |
19.1.322...571.1 |
$x^{19} - 7$ |
$19$ |
[1,9] |
$-\,7^{18}\cdot 19^{19}$ |
$2$ |
$120.05293909$ |
$138.97496464978437$ |
|
|
✓ |
$F_{19}$ (as 19T6) |
trivial |
$2$ |
$9$ |
$4049705166630$ |
19.19.102...168.1 |
$x^{19} - 57 x^{17} + 1368 x^{15} - 17955 x^{13} + 140049 x^{11} - 660231 x^{9} + 1828332 x^{7} - 2742498 x^{5} + 1869885 x^{3} - 373977 x - 67446$ |
$19$ |
[19,0] |
$2^{27}\cdot 3^{18}\cdot 19^{19}$ |
$3$ |
$144.059757112$ |
$176.1456731700741$ |
|
|
? |
$F_{19}$ (as 19T6) |
trivial |
$2$ |
$18$ |
$1170215493390000$ |
19.19.197...000.1 |
$x^{19} - 95 x^{17} + 3800 x^{15} - 83125 x^{13} + 1080625 x^{11} - 8490625 x^{9} + 39187500 x^{7} - 97968750 x^{5} + 111328125 x^{3} - 37109375 x - 8731210$ |
$19$ |
[19,0] |
$2^{18}\cdot 5^{18}\cdot 19^{19}$ |
$3$ |
$168.314690178$ |
$202.08286084610364$ |
|
|
? |
$F_{19}$ (as 19T6) |
trivial |
$2$ |
$18$ |
$5210002514550000$ |