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 |
| 5.4.5.36a1.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 5$ |
not computed |
$ $not computed$ $ |
$4$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.2 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 25 x^{3} ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.3 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 25 x^{2} ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.4 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(50 x^{3} + 25 x^{2}\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.5 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 50 x^{2} ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.6 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 50 x^{2}\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.7 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 75 x^{2} ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.8 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 100 x^{2} ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.9 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 25 x ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.10 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{2} + 25 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.11 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 25 x^{2} + 25 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5^2:C_4^2$ (as 20T91) |
$400$ |
$2$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.12 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(50 x^{2} + 25 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.13 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 50 x^{2} + 25 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.14 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(75 x^{3} + 100 x^{2} + 25 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.15 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 50 x ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.16 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 50 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.17 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{2} + 50 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.18 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(50 x^{3} + 50 x^{2} + 50 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$2$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.19 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 75 x^{2} + 50 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.20 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(100 x^{2} + 50 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.21 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 75 x ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.22 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{2} + 75 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.23 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(50 x^{2} + 75 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.24 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(75 x^{2} + 75 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.25 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 100 x^{2} + 75 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.26 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 100 x ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.27 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(50 x^{3} + 100 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.28 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{2} + 100 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.29 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 25 x^{2} + 100 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.30 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(50 x^{2} + 100 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.31 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 50 x^{2} + 100 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.32 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(75 x^{2} + 100 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.33 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 100 x^{2} + 100 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.34 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 25 ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$4$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.35 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.36 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{2} + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.37 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(50 x^{3} + 25 x^{2} + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.38 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(50 x^{2} + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.39 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 50 x^{2} + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.40 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(75 x^{2} + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.41 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(100 x^{2} + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.42 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.43 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{2} + 25 x + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.44 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 25 x^{2} + 25 x + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$2$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.45 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(50 x^{2} + 25 x + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.46 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 50 x^{2} + 25 x + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$F_5^2$ (as 20T102) |
$400$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.47 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(75 x^{3} + 100 x^{2} + 25 x + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.48 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(50 x + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.49 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{3} + 50 x + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.36a1.50 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(25 x^{2} + 50 x + 25\right) ( x^{4} + 4 x^{2} + 4 x + 2 ) + 5$ |
$C_5:F_5^2$ (as 20T232) |
$2000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[5, 0]$ |
$[1]$ |
$z + (t^3 + 3 t^2 + 3 t + 2)$ |
undefined |
