Defining polynomial
|
\(x^{6} + 214\)
|
Invariants
| Base field: | $\Q_{107}$ |
| Degree $d$: | $6$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $5$ |
| Discriminant root field: | $\Q_{107}(\sqrt{107})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{107})$: | $C_2$ |
| This field is not Galois over $\Q_{107}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $106 = (107 - 1)$ |
Intermediate fields
| $\Q_{107}(\sqrt{107})$, 107.1.3.2a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{107}$ |
| Relative Eisenstein polynomial: |
\( x^{6} + 214 \)
|
Ramification polygon
| Residual polynomials: | $z^5 + 6 z^4 + 15 z^3 + 20 z^2 + 15 z + 6$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $12$ |
| Galois group: | $D_6$ (as 6T3) |
| Inertia group: | $C_6$ (as 6T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $6$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.8333333333333334$ |
| Galois splitting model: | $x^{6} - 107$ |