| $x^{5} + b_{7} \pi^{2} x^{2} + a_{6} \pi^{2} x + \pi$ |
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 |
| 5.1.20.25a1.4 |
$x^{20} + 10 x^{6} + 10$ |
$C_5^2:\OD_{16}$ (as 20T109) |
$400$ |
$2$ |
not computed |
not computed |
not computed |
not computed |
$[6, 0]$ |
$[1, 1]$ |
$z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 4$ |
undefined |
| 5.1.20.25a1.5 |
$x^{20} + 5 x^{7} + 10 x^{6} + 10$ |
$C_5^4.\OD_{16}$ (as 20T386) |
$10000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[6, 0]$ |
$[1, 1]$ |
$z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 4$ |
undefined |
| 5.1.20.25a1.6 |
$x^{20} + 10 x^{7} + 10 x^{6} + 10$ |
$C_5^4.\OD_{16}$ (as 20T386) |
$10000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[6, 0]$ |
$[1, 1]$ |
$z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 4$ |
undefined |
| 5.1.20.25a2.4 |
$x^{20} + 5 x^{6} + 10$ |
not computed |
$ $not computed$ $ |
$2$ |
not computed |
not computed |
not computed |
not computed |
$[6, 0]$ |
$[1, 2]$ |
$z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 2$ |
undefined |
| 5.1.20.25a2.5 |
$x^{20} + 5 x^{7} + 5 x^{6} + 10$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[6, 0]$ |
$[1, 2]$ |
$z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 2$ |
undefined |
| 5.1.20.25a2.6 |
$x^{20} + 10 x^{7} + 5 x^{6} + 10$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[6, 0]$ |
$[1, 2]$ |
$z^{15} + 4 z^{10} + z^5 + 4,4 z^2 + 2$ |
undefined |
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