| $x^{5} + a_{2} \pi x^{2} + \pi$ |
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 |
| 5.1.15.16a1.1 |
$x^{15} + 10 x^{2} + 5$ |
$(C_5^2 : C_3):C_4$ (as 15T17) |
$300$ |
$1$ |
$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ |
$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ |
$[\frac{7}{6}]^{2}_{2}$ |
$[\frac{1}{6}]^{2}_{2}$ |
$[2, 0]$ |
$[2, 2]$ |
$z^{10} + 3 z^5 + 3,3 z^2 + 1$ |
undefined |
| 5.1.15.16a1.2 |
$x^{15} + 15 x^{2} + 5$ |
$(C_5^2 : C_3):C_4$ (as 15T17) |
$300$ |
$1$ |
$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ |
$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ |
$[\frac{7}{6}]^{2}_{2}$ |
$[\frac{1}{6}]^{2}_{2}$ |
$[2, 0]$ |
$[2, 2]$ |
$z^{10} + 3 z^5 + 3,3 z^2 + 4$ |
undefined |
| 5.1.15.16a2.1 |
$x^{15} + 5 x^{2} + 5$ |
$((C_5^2 : C_3):C_2):C_2$ (as 15T18) |
$300$ |
$1$ |
$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ |
$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ |
$[\frac{7}{6}]^{2}_{2}$ |
$[\frac{1}{6}]^{2}_{2}$ |
$[2, 0]$ |
$[2, 1]$ |
$z^{10} + 3 z^5 + 3,3 z^2 + 3$ |
undefined |
| 5.1.15.16a2.2 |
$x^{15} + 20 x^{2} + 5$ |
$((C_5^2 : C_3):C_2):C_2$ (as 15T18) |
$300$ |
$1$ |
$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$ |
$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$ |
$[\frac{7}{6}]^{2}_{2}$ |
$[\frac{1}{6}]^{2}_{2}$ |
$[2, 0]$ |
$[2, 1]$ |
$z^{10} + 3 z^5 + 3,3 z^2 + 2$ |
undefined |
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