Defining polynomial
| $x^{8} + 19d_{0}$ |
Invariants
| Residue field characteristic: | $19$ |
| Degree: | $8$ |
| Base field: | $\Q_{19}(\sqrt{2})$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $7$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $5$ (complete) |
| Ambiguity: | $8$ |
| Mass: | $1$ |
| Absolute Mass: | $1/2$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 19 }$ within this relative family, not the relative extension.
| Galois group: | $QD_{16}$ (show 2), $C_8.C_4$ (show 1), $C_8.C_8$ (show 2) |
| Hidden Artin slopes: | $[\ ]$ (show 2), $[\ ]^{2}$ (show 1), $[\ ]^{4}$ (show 2) |
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | undefined |
Fields
Showing all 5
Download displayed columns for results| Label | Polynomial $/ \Q_p$ | Galois group $/ \Q_p$ | Galois degree $/ \Q_p$ | $\#\Aut(K/\Q_p)$ | Hidden Artin slopes $/ \Q_p$ | Ind. of Insep. $/ \Q_p$ | Assoc. Inertia $/ \Q_p$ | Jump Set |
|---|---|---|---|---|---|---|---|---|
| 19.2.8.14a1.1 | $( x^{2} + 18 x + 2 )^{8} + 19 x$ | $C_8.C_8$ (as 16T124) | $64$ | $8$ | $[\ ]^{4}$ | $[0]$ | $[1]$ | undefined |
| 19.2.8.14a1.2 | $( x^{2} + 18 x + 2 )^{8} + 19$ | $QD_{16}$ (as 16T12) | $16$ | $16$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 19.2.8.14a1.3 | $( x^{2} + 18 x + 2 )^{8} + 304 x + 38$ | $QD_{16}$ (as 16T12) | $16$ | $16$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 19.2.8.14a1.4 | $( x^{2} + 18 x + 2 )^{8} + 133 x + 171$ | $C_8.C_8$ (as 16T124) | $64$ | $8$ | $[\ ]^{4}$ | $[0]$ | $[1]$ | undefined |
| 19.2.8.14a1.5 | $( x^{2} + 18 x + 2 )^{8} + 19 x + 323$ | $C_8.C_4$ (as 16T49) | $32$ | $8$ | $[\ ]^{2}$ | $[0]$ | $[1]$ | undefined |