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 |
| 3.1.3.3a1.1 |
$3$ |
$x^{3} + 3 x + 3$ |
$3$ |
$1$ |
$3$ |
$3$ |
$S_3$ (as 3T2) |
$1$ |
$2$ |
$[\frac{3}{2}]$ |
$[\frac{1}{2}]$ |
$[\frac{3}{2}]_{2}$ |
$[\frac{1}{2}]_{2}$ |
$[\ ]_{2}$ |
$[\ ]_{2}$ |
$1$ |
$t + 1$ |
$x^{3} + 3 x + 3$ |
$[1, 0]$ |
$[1]$ |
$z + 2$ |
undefined |
| 3.1.3.3a1.2 |
$3$ |
$x^{3} + 6 x + 3$ |
$3$ |
$1$ |
$3$ |
$3$ |
$S_3$ (as 3T2) |
$1$ |
$2$ |
$[\frac{3}{2}]$ |
$[\frac{1}{2}]$ |
$[\frac{3}{2}]_{2}$ |
$[\frac{1}{2}]_{2}$ |
$[\ ]_{2}$ |
$[\ ]_{2}$ |
$1$ |
$t + 1$ |
$x^{3} + 6 x + 3$ |
$[1, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 3.1.3.4a1.1 |
$3$ |
$x^{3} + 3 x^{2} + 3$ |
$3$ |
$1$ |
$3$ |
$4$ |
$S_3$ (as 3T2) |
$2$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]^{2}$ |
$[1]^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$1$ |
$t + 1$ |
$x^{3} + 3 x^{2} + 3$ |
$[2, 0]$ |
$[2]$ |
$z^2 + 1$ |
undefined |
| 3.1.3.4a2.1 |
$3$ |
$x^{3} + 6 x^{2} + 3$ |
$3$ |
$1$ |
$3$ |
$4$ |
$C_3$ (as 3T1) |
$1$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]$ |
$[1]$ |
$[\ ]$ |
$[\ ]$ |
$3$ |
$t + 1$ |
$x^{3} + 6 x^{2} + 3$ |
$[2, 0]$ |
$[1]$ |
$z^2 + 2$ |
undefined |
| 3.1.3.4a2.2 |
$3$ |
$x^{3} + 6 x^{2} + 12$ |
$3$ |
$1$ |
$3$ |
$4$ |
$C_3$ (as 3T1) |
$1$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]$ |
$[1]$ |
$[\ ]$ |
$[\ ]$ |
$3$ |
$t + 1$ |
$x^{3} + 6 x^{2} + 12$ |
$[2, 0]$ |
$[1]$ |
$z^2 + 2$ |
undefined |
| 3.1.3.4a2.3 |
$3$ |
$x^{3} + 6 x^{2} + 21$ |
$3$ |
$1$ |
$3$ |
$4$ |
$C_3$ (as 3T1) |
$1$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]$ |
$[1]$ |
$[\ ]$ |
$[\ ]$ |
$3$ |
$t + 1$ |
$x^{3} + 6 x^{2} + 21$ |
$[2, 0]$ |
$[1]$ |
$z^2 + 2$ |
undefined |
| 3.1.3.5a1.1 |
$3$ |
$x^{3} + 3$ |
$3$ |
$1$ |
$3$ |
$5$ |
$S_3$ (as 3T2) |
$1$ |
$2$ |
$[\frac{5}{2}]$ |
$[\frac{3}{2}]$ |
$[\frac{5}{2}]_{2}$ |
$[\frac{3}{2}]_{2}$ |
$[\ ]_{2}$ |
$[\ ]_{2}$ |
$1$ |
$t + 1$ |
$x^{3} + 3$ |
$[3, 0]$ |
$[1]$ |
$z + 2$ |
undefined |
| 3.1.3.5a1.2 |
$3$ |
$x^{3} + 9 x + 3$ |
$3$ |
$1$ |
$3$ |
$5$ |
$S_3$ (as 3T2) |
$1$ |
$2$ |
$[\frac{5}{2}]$ |
$[\frac{3}{2}]$ |
$[\frac{5}{2}]_{2}$ |
$[\frac{3}{2}]_{2}$ |
$[\ ]_{2}$ |
$[\ ]_{2}$ |
$1$ |
$t + 1$ |
$x^{3} + 9 x + 3$ |
$[3, 0]$ |
$[1]$ |
$z + 2$ |
undefined |
| 3.1.3.5a1.3 |
$3$ |
$x^{3} + 18 x + 3$ |
$3$ |
$1$ |
$3$ |
$5$ |
$S_3$ (as 3T2) |
$1$ |
$2$ |
$[\frac{5}{2}]$ |
$[\frac{3}{2}]$ |
$[\frac{5}{2}]_{2}$ |
$[\frac{3}{2}]_{2}$ |
$[\ ]_{2}$ |
$[\ ]_{2}$ |
$1$ |
$t + 1$ |
$x^{3} + 18 x + 3$ |
$[3, 0]$ |
$[1]$ |
$z + 2$ |
undefined |
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