Defining polynomial
|
\(x^{14} + 22\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $13$ |
| Discriminant root field: | $\Q_{11}(\sqrt{11})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 11 }) }$: | $2$ |
| This field is not Galois over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{11}(\sqrt{11})$, 11.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 22 \)
|
Ramification polygon
| Residual polynomials: | $z^{13} + 3z^{12} + 3z^{11} + z^{10} + z^{2} + 3z + 3$ |
| Associated inertia: | $3$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_7:C_6$ (as 14T5) |
| Inertia group: | $C_{14}$ (as 14T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $3$ |
| Tame degree: | $14$ |
| Wild slopes: | None |
| Galois mean slope: | $13/14$ |
| Galois splitting model: | Not computed |