Defining polynomial over unramified subextension
| $x^{8} + 131d_{0}$ |
Invariants
| Residue field characteristic: | $131$ |
| Degree: | $16$ |
| Base field: | $\Q_{131}$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $14$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $5$ (complete) |
| Ambiguity: | $16$ |
| Mass: | $1$ |
| Absolute Mass: | $1/2$ |
Varying
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | undefined |
Galois groups and Hidden Artin slopes
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Fields
Showing all 5
Download displayed columns for results| Label | Polynomial | Galois group | Galois degree | $\#\Aut(K/\Q_p)$ | Hidden Artin slopes | Ind. of Insep. | Assoc. Inertia | Jump Set |
|---|---|---|---|---|---|---|---|---|
| 131.2.8.14a1.1 | $( x^{2} + 127 x + 2 )^{8} + 131 x$ | $C_8.C_8$ (as 16T124) | $64$ | $8$ | $[\ ]^{4}$ | $[0]$ | $[1]$ | undefined |
| 131.2.8.14a1.2 | $( x^{2} + 127 x + 2 )^{8} + 131$ | $QD_{16}$ (as 16T12) | $16$ | $16$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 131.2.8.14a1.3 | $( x^{2} + 127 x + 2 )^{8} + 4323 x + 4585$ | $C_8.C_8$ (as 16T124) | $64$ | $8$ | $[\ ]^{4}$ | $[0]$ | $[1]$ | undefined |
| 131.2.8.14a1.4 | $( x^{2} + 127 x + 2 )^{8} + 6288 x + 13493$ | $QD_{16}$ (as 16T12) | $16$ | $16$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 131.2.8.14a1.5 | $( x^{2} + 127 x + 2 )^{8} + 524 x + 16899$ | $C_8.C_4$ (as 16T49) | $32$ | $8$ | $[\ ]^{2}$ | $[0]$ | $[1]$ | undefined |