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 |
| 5.5.1.0a1.1 |
$5$ |
$x^{5} + 4 x + 3$ |
$5$ |
$5$ |
$1$ |
$0$ |
$C_5$ (as 5T1) |
$5$ |
$1$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]^{5}$ |
$[\ ]^{5}$ |
$[\ ]$ |
$[\ ]$ |
$5$ |
$t^{5} + 4 t + 3$ |
$x - 5$ |
$[0]$ |
$[\ ]$ |
|
undefined |
| 5.1.5.5a1.1 |
$5$ |
$x^{5} + 5 x + 5$ |
$5$ |
$1$ |
$5$ |
$5$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{5}{4}]$ |
$[\frac{1}{4}]$ |
$[\frac{5}{4}]_{4}$ |
$[\frac{1}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 5 x + 5$ |
$[1, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.5a1.2 |
$5$ |
$x^{5} + 10 x + 5$ |
$5$ |
$1$ |
$5$ |
$5$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{5}{4}]$ |
$[\frac{1}{4}]$ |
$[\frac{5}{4}]_{4}$ |
$[\frac{1}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 10 x + 5$ |
$[1, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.5a1.3 |
$5$ |
$x^{5} + 15 x + 5$ |
$5$ |
$1$ |
$5$ |
$5$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{5}{4}]$ |
$[\frac{1}{4}]$ |
$[\frac{5}{4}]_{4}$ |
$[\frac{1}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 15 x + 5$ |
$[1, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.5a1.4 |
$5$ |
$x^{5} + 20 x + 5$ |
$5$ |
$1$ |
$5$ |
$5$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{5}{4}]$ |
$[\frac{1}{4}]$ |
$[\frac{5}{4}]_{4}$ |
$[\frac{1}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 20 x + 5$ |
$[1, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.6a1.1 |
$5$ |
$x^{5} + 10 x^{2} + 5$ |
$5$ |
$1$ |
$5$ |
$6$ |
$D_{5}$ (as 5T2) |
$1$ |
$2$ |
$[\frac{3}{2}]$ |
$[\frac{1}{2}]$ |
$[\frac{3}{2}]_{2}$ |
$[\frac{1}{2}]_{2}$ |
$[\ ]_{2}$ |
$[\ ]_{2}$ |
$1$ |
$t + 3$ |
$x^{5} + 10 x^{2} + 5$ |
$[2, 0]$ |
$[1]$ |
$z^2 + 1$ |
undefined |
| 5.1.5.6a1.2 |
$5$ |
$x^{5} + 15 x^{2} + 5$ |
$5$ |
$1$ |
$5$ |
$6$ |
$D_{5}$ (as 5T2) |
$1$ |
$2$ |
$[\frac{3}{2}]$ |
$[\frac{1}{2}]$ |
$[\frac{3}{2}]_{2}$ |
$[\frac{1}{2}]_{2}$ |
$[\ ]_{2}$ |
$[\ ]_{2}$ |
$1$ |
$t + 3$ |
$x^{5} + 15 x^{2} + 5$ |
$[2, 0]$ |
$[1]$ |
$z^2 + 1$ |
undefined |
| 5.1.5.6a2.1 |
$5$ |
$x^{5} + 5 x^{2} + 5$ |
$5$ |
$1$ |
$5$ |
$6$ |
$F_5$ (as 5T3) |
$2$ |
$2$ |
$[\frac{3}{2}]$ |
$[\frac{1}{2}]$ |
$[\frac{3}{2}]_{2}^{2}$ |
$[\frac{1}{2}]_{2}^{2}$ |
$[\ ]^{2}_{2}$ |
$[\ ]^{2}_{2}$ |
$1$ |
$t + 3$ |
$x^{5} + 5 x^{2} + 5$ |
$[2, 0]$ |
$[2]$ |
$z^2 + 2$ |
undefined |
| 5.1.5.6a2.2 |
$5$ |
$x^{5} + 20 x^{2} + 5$ |
$5$ |
$1$ |
$5$ |
$6$ |
$F_5$ (as 5T3) |
$2$ |
$2$ |
$[\frac{3}{2}]$ |
$[\frac{1}{2}]$ |
$[\frac{3}{2}]_{2}^{2}$ |
$[\frac{1}{2}]_{2}^{2}$ |
$[\ ]^{2}_{2}$ |
$[\ ]^{2}_{2}$ |
$1$ |
$t + 3$ |
$x^{5} + 20 x^{2} + 5$ |
$[2, 0]$ |
$[2]$ |
$z^2 + 2$ |
undefined |
| 5.1.5.7a1.1 |
$5$ |
$x^{5} + 5 x^{3} + 5$ |
$5$ |
$1$ |
$5$ |
$7$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{7}{4}]_{4}$ |
$[\frac{3}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 5 x^{3} + 5$ |
$[3, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.7a1.2 |
$5$ |
$x^{5} + 10 x^{3} + 5$ |
$5$ |
$1$ |
$5$ |
$7$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{7}{4}]_{4}$ |
$[\frac{3}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 10 x^{3} + 5$ |
$[3, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.7a1.3 |
$5$ |
$x^{5} + 15 x^{3} + 5$ |
$5$ |
$1$ |
$5$ |
$7$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{7}{4}]_{4}$ |
$[\frac{3}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 15 x^{3} + 5$ |
$[3, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.7a1.4 |
$5$ |
$x^{5} + 20 x^{3} + 5$ |
$5$ |
$1$ |
$5$ |
$7$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{7}{4}]_{4}$ |
$[\frac{3}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 20 x^{3} + 5$ |
$[3, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.8a1.1 |
$5$ |
$x^{5} + 5 x^{4} + 5$ |
$5$ |
$1$ |
$5$ |
$8$ |
$D_{5}$ (as 5T2) |
$2$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]^{2}$ |
$[1]^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$1$ |
$t + 3$ |
$x^{5} + 5 x^{4} + 5$ |
$[4, 0]$ |
$[2]$ |
$z^4 + 1$ |
undefined |
| 5.1.5.8a2.1 |
$5$ |
$x^{5} + 10 x^{4} + 5$ |
$5$ |
$1$ |
$5$ |
$8$ |
$F_5$ (as 5T3) |
$4$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]^{4}$ |
$[1]^{4}$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 10 x^{4} + 5$ |
$[4, 0]$ |
$[4]$ |
$z^4 + 2$ |
undefined |
| 5.1.5.8a3.1 |
$5$ |
$x^{5} + 15 x^{4} + 5$ |
$5$ |
$1$ |
$5$ |
$8$ |
$F_5$ (as 5T3) |
$4$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]^{4}$ |
$[1]^{4}$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 15 x^{4} + 5$ |
$[4, 0]$ |
$[4]$ |
$z^4 + 3$ |
undefined |
| 5.1.5.8a4.1 |
$5$ |
$x^{5} + 20 x^{4} + 5$ |
$5$ |
$1$ |
$5$ |
$8$ |
$C_5$ (as 5T1) |
$1$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]$ |
$[1]$ |
$[\ ]$ |
$[\ ]$ |
$5$ |
$t + 3$ |
$x^{5} + 20 x^{4} + 5$ |
$[4, 0]$ |
$[1]$ |
$z^4 + 4$ |
undefined |
| 5.1.5.8a4.2 |
$5$ |
$x^{5} + 20 x^{4} + 30$ |
$5$ |
$1$ |
$5$ |
$8$ |
$C_5$ (as 5T1) |
$1$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]$ |
$[1]$ |
$[\ ]$ |
$[\ ]$ |
$5$ |
$t + 3$ |
$x^{5} + 20 x^{4} + 30$ |
$[4, 0]$ |
$[1]$ |
$z^4 + 4$ |
undefined |
| 5.1.5.8a4.3 |
$5$ |
$x^{5} + 20 x^{4} + 55$ |
$5$ |
$1$ |
$5$ |
$8$ |
$C_5$ (as 5T1) |
$1$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]$ |
$[1]$ |
$[\ ]$ |
$[\ ]$ |
$5$ |
$t + 3$ |
$x^{5} + 20 x^{4} + 55$ |
$[4, 0]$ |
$[1]$ |
$z^4 + 4$ |
undefined |
| 5.1.5.8a4.4 |
$5$ |
$x^{5} + 20 x^{4} + 80$ |
$5$ |
$1$ |
$5$ |
$8$ |
$C_5$ (as 5T1) |
$1$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]$ |
$[1]$ |
$[\ ]$ |
$[\ ]$ |
$5$ |
$t + 3$ |
$x^{5} + 20 x^{4} + 80$ |
$[4, 0]$ |
$[1]$ |
$z^4 + 4$ |
undefined |
| 5.1.5.8a4.5 |
$5$ |
$x^{5} + 20 x^{4} + 105$ |
$5$ |
$1$ |
$5$ |
$8$ |
$C_5$ (as 5T1) |
$1$ |
$1$ |
$[2]$ |
$[1]$ |
$[2]$ |
$[1]$ |
$[\ ]$ |
$[\ ]$ |
$5$ |
$t + 3$ |
$x^{5} + 20 x^{4} + 105$ |
$[4, 0]$ |
$[1]$ |
$z^4 + 4$ |
undefined |
| 5.1.5.9a1.1 |
$5$ |
$x^{5} + 5$ |
$5$ |
$1$ |
$5$ |
$9$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{9}{4}]_{4}$ |
$[\frac{5}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 5$ |
$[5, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.9a1.2 |
$5$ |
$x^{5} + 25 x + 5$ |
$5$ |
$1$ |
$5$ |
$9$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{9}{4}]_{4}$ |
$[\frac{5}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 25 x + 5$ |
$[5, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.9a1.3 |
$5$ |
$x^{5} + 50 x + 5$ |
$5$ |
$1$ |
$5$ |
$9$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{9}{4}]_{4}$ |
$[\frac{5}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 50 x + 5$ |
$[5, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.9a1.4 |
$5$ |
$x^{5} + 75 x + 5$ |
$5$ |
$1$ |
$5$ |
$9$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{9}{4}]_{4}$ |
$[\frac{5}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 75 x + 5$ |
$[5, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
| 5.1.5.9a1.5 |
$5$ |
$x^{5} + 100 x + 5$ |
$5$ |
$1$ |
$5$ |
$9$ |
$F_5$ (as 5T3) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{9}{4}]_{4}$ |
$[\frac{5}{4}]_{4}$ |
$[\ ]_{4}$ |
$[\ ]_{4}$ |
$1$ |
$t + 3$ |
$x^{5} + 100 x + 5$ |
$[5, 0]$ |
$[1]$ |
$z + 1$ |
undefined |
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