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 |
| 5.4.5.32a1.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 5 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_5:C_4$ (as 20T2) |
$20$ |
$20$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (3 t^3 + t^2 + t + 2)$ |
undefined |
| 5.4.5.32a1.2 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 5 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 25 x + 5$ |
$C_5:C_{20}$ (as 20T25) |
$100$ |
$10$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (3 t^3 + t^2 + t + 2)$ |
undefined |
| 5.4.5.32a1.3 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 5 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 50 x + 5$ |
$C_5:C_{20}$ (as 20T25) |
$100$ |
$10$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (3 t^3 + t^2 + t + 2)$ |
undefined |
| 5.4.5.32a2.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 10 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
not computed |
$ $not computed$ $ |
$20$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (t^3 + 2 t^2 + 2 t + 4)$ |
undefined |
| 5.4.5.32a2.2 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 10 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 25 x^{2} + 5$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (t^3 + 2 t^2 + 2 t + 4)$ |
undefined |
| 5.4.5.32a3.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 15 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$F_5$ (as 20T5) |
$20$ |
$20$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 3 t^2 + 3 t + 1)$ |
undefined |
| 5.4.5.32a3.2 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 15 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 25 x + 5$ |
$C_5\times F_5$ (as 20T29) |
$100$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 3 t^2 + 3 t + 1)$ |
undefined |
| 5.4.5.32a4.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 20 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_{20}$ (as 20T1) |
$20$ |
$20$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (2 t^3 + 4 t^2 + 4 t + 3)$ |
undefined |
| 5.4.5.32a4.2 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 20 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 30$ |
$C_{20}$ (as 20T1) |
$20$ |
$20$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (2 t^3 + 4 t^2 + 4 t + 3)$ |
undefined |
| 5.4.5.32a4.3 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 20 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 55$ |
not computed |
$ $not computed$ $ |
$20$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (2 t^3 + 4 t^2 + 4 t + 3)$ |
undefined |
| 5.4.5.32a4.4 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 20 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 80$ |
$C_{20}$ (as 20T1) |
$20$ |
$20$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (2 t^3 + 4 t^2 + 4 t + 3)$ |
undefined |
| 5.4.5.32a4.5 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 20 ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 105$ |
not computed |
$ $not computed$ $ |
$20$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (2 t^3 + 4 t^2 + 4 t + 3)$ |
undefined |
| 5.4.5.32a5.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 5 x ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[4]$ |
$z^4 + (t^3 + 4 t^2 + 4)$ |
undefined |
| 5.4.5.32a6.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x + 5\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[4]$ |
$z^4 + (4 t^3 + t + 1)$ |
undefined |
| 5.4.5.32a7.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 10 x ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_5^4:C_{16}$ (as 20T392) |
$10000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[4]$ |
$z^4 + (2 t^3 + 3 t^2 + 3)$ |
undefined |
| 5.4.5.32a8.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 15 x ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[4]$ |
$z^4 + (3 t^3 + 2 t^2 + 2)$ |
undefined |
| 5.4.5.32a9.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 5\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (t^3 + 3 t^2 + t + 4)$ |
undefined |
| 5.4.5.32a9.2 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 5\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 30$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (t^3 + 3 t^2 + t + 4)$ |
undefined |
| 5.4.5.32a9.3 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 5\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 55$ |
$C_5^2:C_{20}$ (as 20T127) |
$500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (t^3 + 3 t^2 + t + 4)$ |
undefined |
| 5.4.5.32a9.4 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 5\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 80$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (t^3 + 3 t^2 + t + 4)$ |
undefined |
| 5.4.5.32a9.5 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 5\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 105$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (t^3 + 3 t^2 + t + 4)$ |
undefined |
| 5.4.5.32a10.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 10\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 4 t^2 + 2 t + 1)$ |
undefined |
| 5.4.5.32a10.2 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 10\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 30$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 4 t^2 + 2 t + 1)$ |
undefined |
| 5.4.5.32a10.3 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 10\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 55$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 4 t^2 + 2 t + 1)$ |
undefined |
| 5.4.5.32a10.4 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 10\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 80$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 4 t^2 + 2 t + 1)$ |
undefined |
| 5.4.5.32a10.5 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 10\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 105$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 4 t^2 + 2 t + 1)$ |
undefined |
| 5.4.5.32a11.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(15 x + 20\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_5^4:C_{16}$ (as 20T392) |
$10000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[4]$ |
$z^4 + (t^2 + 4 t)$ |
undefined |
| 5.4.5.32a12.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 20 x ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_5^4:C_{16}$ (as 20T392) |
$10000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[4]$ |
$z^4 + (4 t^3 + t^2 + 1)$ |
undefined |
| 5.4.5.32a13.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(20 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_5^3:C_4$ (as 20T125) |
$500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (3 t^3 + 4 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.32a13.2 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(20 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 25 x^{2} + 5$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (3 t^3 + 4 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.32a13.3 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(20 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 50 x^{2} + 5$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (3 t^3 + 4 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.32a13.4 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(20 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 75 x^{2} + 5$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (3 t^3 + 4 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.32a13.5 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(20 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 100 x^{2} + 5$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (3 t^3 + 4 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.32a14.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 5 x^{2} ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[2]$ |
$z^4 + (4 t^3 + t^2 + 3)$ |
undefined |
| 5.4.5.32a15.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[4]$ |
$z^4 + (2 t^3 + 2 t^2 + t)$ |
undefined |
| 5.4.5.32a16.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + 2$ |
undefined |
| 5.4.5.32a16.2 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 30$ |
$C_5^2:C_{20}$ (as 20T127) |
$500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + 2$ |
undefined |
| 5.4.5.32a16.3 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 55$ |
$C_5^2:C_{20}$ (as 20T127) |
$500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + 2$ |
undefined |
| 5.4.5.32a16.4 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 80$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + 2$ |
undefined |
| 5.4.5.32a16.5 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5 x\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 105$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + 2$ |
undefined |
| 5.4.5.32a17.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 3 t^2 + 3 t + 3)$ |
undefined |
| 5.4.5.32a17.2 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 30$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 3 t^2 + 3 t + 3)$ |
undefined |
| 5.4.5.32a17.3 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 55$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 3 t^2 + 3 t + 3)$ |
undefined |
| 5.4.5.32a17.4 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 80$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 3 t^2 + 3 t + 3)$ |
undefined |
| 5.4.5.32a17.5 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 5 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 105$ |
$C_5\wr C_4$ (as 20T237) |
$2500$ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (4 t^3 + 3 t^2 + 3 t + 3)$ |
undefined |
| 5.4.5.32a18.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 10 x + 5\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
not computed |
$ $not computed$ $ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[2]$ |
$z^4 + (4 t^3 + t + 3)$ |
undefined |
| 5.4.5.32a19.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 10 x + 15\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_5^4:C_8$ (as 20T297) |
$5000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[2]$ |
$z^4 + (2 t^2 + 3 t + 2)$ |
undefined |
| 5.4.5.32a20.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(5 x^{2} + 15 x + 20\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_5^4:C_{16}$ (as 20T392) |
$10000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[4]$ |
$z^4 + (4 t^3 + 2 t^2 + 4 t + 3)$ |
undefined |
| 5.4.5.32a21.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + 10 x^{2} ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$C_5^4:C_8$ (as 20T297) |
$5000$ |
$1$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[2]$ |
$z^4 + (3 t^3 + 2 t^2 + 1)$ |
undefined |
| 5.4.5.32a22.1 |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{5} + \left(10 x^{2} + 20\right) ( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
not computed |
$ $not computed$ $ |
$5$ |
not computed |
not computed |
not computed |
not computed |
$[4, 0]$ |
$[1]$ |
$z^4 + (t^2 + 4 t + 4)$ |
undefined |
