The results below are complete, since the LMFDB contains all p-adic fields of degree at most 23 and residue characteristic at most 199
| Label |
$n$ |
Polynomial |
$p$ |
$f$ |
$e$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible Artin slopes |
Visible Swan slopes |
Artin slope content |
Swan slope content |
Hidden Artin slopes |
Hidden Swan slopes |
$\#\Aut(K/\Q_p)$ |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
Resid. Poly |
Jump Set |
| 2.4.1.0a1.1 |
$4$ |
$x^{4} + x + 1$ |
$2$ |
$4$ |
$1$ |
$0$ |
$C_4$ (as 4T1) |
$4$ |
$1$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t^{4} + t + 1$ |
$x - 2$ |
$[0]$ |
$[\ ]$ |
|
$[1]$ |
| 2.2.2.4a1.2 |
$4$ |
$( x^{2} + x + 1 )^{2} + 2 ( x^{2} + x + 1 ) + 4 x + 2$ |
$2$ |
$2$ |
$2$ |
$4$ |
$C_4$ (as 4T1) |
$2$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]^{2}$ |
$[1]^{2}$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t^{2} + t + 1$ |
$x^{2} + 2 x + 4 t + 2$ |
$[1, 0]$ |
$[1]$ |
$z + 1$ |
$[1, 4]$ |
| 2.2.2.6a1.2 |
$4$ |
$( x^{2} + x + 1 )^{2} + 8 x + 2$ |
$2$ |
$2$ |
$2$ |
$6$ |
$C_4$ (as 4T1) |
$2$ |
$1$ |
$[3]$ |
$[2]$ |
$[3]^{2}$ |
$[2]^{2}$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t^{2} + t + 1$ |
$x^{2} + 8 t + 2$ |
$[2, 0]$ |
$[1]$ |
$z + 1$ |
$[1, 3]$ |
| 2.2.2.6a1.6 |
$4$ |
$( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 8 x + 2$ |
$2$ |
$2$ |
$2$ |
$6$ |
$C_4$ (as 4T1) |
$2$ |
$1$ |
$[3]$ |
$[2]$ |
$[3]^{2}$ |
$[2]^{2}$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t^{2} + t + 1$ |
$x^{2} + 4 x + 8 t + 2$ |
$[2, 0]$ |
$[1]$ |
$z + 1$ |
$[1, 3]$ |
| 2.1.4.11a1.9 |
$4$ |
$x^{4} + 4 x^{2} + 2$ |
$2$ |
$1$ |
$4$ |
$11$ |
$C_4$ (as 4T1) |
$1$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[3, 4]$ |
$[2,3]$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t + 1$ |
$x^{4} + 4 x^{2} + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.1.4.11a1.10 |
$4$ |
$x^{4} + 4 x^{2} + 18$ |
$2$ |
$1$ |
$4$ |
$11$ |
$C_4$ (as 4T1) |
$1$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[3, 4]$ |
$[2,3]$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t + 1$ |
$x^{4} + 4 x^{2} + 18$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.1.4.11a1.11 |
$4$ |
$x^{4} + 8 x^{3} + 4 x^{2} + 2$ |
$2$ |
$1$ |
$4$ |
$11$ |
$C_4$ (as 4T1) |
$1$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[3, 4]$ |
$[2,3]$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t + 1$ |
$x^{4} + 8 x^{3} + 4 x^{2} + 2$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.1.4.11a1.12 |
$4$ |
$x^{4} + 8 x^{3} + 4 x^{2} + 18$ |
$2$ |
$1$ |
$4$ |
$11$ |
$C_4$ (as 4T1) |
$1$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[3, 4]$ |
$[2,3]$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t + 1$ |
$x^{4} + 8 x^{3} + 4 x^{2} + 18$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.1.4.11a1.15 |
$4$ |
$x^{4} + 4 x^{2} + 10$ |
$2$ |
$1$ |
$4$ |
$11$ |
$C_4$ (as 4T1) |
$1$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[3, 4]$ |
$[2,3]$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t + 1$ |
$x^{4} + 4 x^{2} + 10$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.1.4.11a1.16 |
$4$ |
$x^{4} + 4 x^{2} + 26$ |
$2$ |
$1$ |
$4$ |
$11$ |
$C_4$ (as 4T1) |
$1$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[3, 4]$ |
$[2,3]$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t + 1$ |
$x^{4} + 4 x^{2} + 26$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.1.4.11a1.17 |
$4$ |
$x^{4} + 8 x^{3} + 4 x^{2} + 10$ |
$2$ |
$1$ |
$4$ |
$11$ |
$C_4$ (as 4T1) |
$1$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[3, 4]$ |
$[2,3]$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t + 1$ |
$x^{4} + 8 x^{3} + 4 x^{2} + 10$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.1.4.11a1.18 |
$4$ |
$x^{4} + 8 x^{3} + 4 x^{2} + 26$ |
$2$ |
$1$ |
$4$ |
$11$ |
$C_4$ (as 4T1) |
$1$ |
$1$ |
$[3, 4]$ |
$[2,3]$ |
$[3, 4]$ |
$[2,3]$ |
$[\ ]$ |
$[\ ]$ |
$4$ |
$t + 1$ |
$x^{4} + 8 x^{3} + 4 x^{2} + 26$ |
$[8, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
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