| $x^{5} + a_{3} \pi x^{3} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 5 }$ within this relative family, not the relative extension.
| Label |
Polynomial $/ \Q_p$ |
Galois group $/ \Q_p$ |
Galois degree $/ \Q_p$ |
$\#\Aut(K/\Q_p)$ |
Artin slope content $/ \Q_p$ |
Swan slope content $/ \Q_p$ |
Hidden Artin slopes $/ \Q_p$ |
Hidden Swan slopes $/ \Q_p$ |
Ind. of Insep. $/ \Q_p$ |
Assoc. Inertia $/ \Q_p$ |
Resid. Poly |
Jump Set |
| 5.1.15.17a1.1 |
$x^{15} + 5 x^{3} + 5$ |
$F_5 \times S_3$ (as 15T11) |
$120$ |
$1$ |
$[\frac{5}{4}]_{12}^{2}$ |
$[\frac{1}{4}]_{12}^{2}$ |
$[\ ]^{2}_{4}$ |
$[\ ]^{2}_{4}$ |
$[3, 0]$ |
$[2, 1]$ |
$z^{10} + 3 z^5 + 3,3 z + 2$ |
undefined |
| 5.1.15.17a1.2 |
$x^{15} + 10 x^{3} + 5$ |
$F_5 \times S_3$ (as 15T11) |
$120$ |
$1$ |
$[\frac{5}{4}]_{12}^{2}$ |
$[\frac{1}{4}]_{12}^{2}$ |
$[\ ]^{2}_{4}$ |
$[\ ]^{2}_{4}$ |
$[3, 0]$ |
$[2, 1]$ |
$z^{10} + 3 z^5 + 3,3 z + 4$ |
undefined |
| 5.1.15.17a1.3 |
$x^{15} + 15 x^{3} + 5$ |
$F_5 \times S_3$ (as 15T11) |
$120$ |
$1$ |
$[\frac{5}{4}]_{12}^{2}$ |
$[\frac{1}{4}]_{12}^{2}$ |
$[\ ]^{2}_{4}$ |
$[\ ]^{2}_{4}$ |
$[3, 0]$ |
$[2, 1]$ |
$z^{10} + 3 z^5 + 3,3 z + 1$ |
undefined |
| 5.1.15.17a1.4 |
$x^{15} + 20 x^{3} + 5$ |
$F_5 \times S_3$ (as 15T11) |
$120$ |
$1$ |
$[\frac{5}{4}]_{12}^{2}$ |
$[\frac{1}{4}]_{12}^{2}$ |
$[\ ]^{2}_{4}$ |
$[\ ]^{2}_{4}$ |
$[3, 0]$ |
$[2, 1]$ |
$z^{10} + 3 z^5 + 3,3 z + 3$ |
undefined |
Download
displayed columns for
results
