Defining polynomial
|
\(x^{12} + 42\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $11$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{7})$: | $C_6$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $6 = (7 - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{7})$, 7.1.3.2a1.1, 7.1.4.3a1.2, 7.1.6.5a1.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 42 \)
|
Ramification polygon
| Residual polynomials: | $z^{11} + 5 z^{10} + 3 z^9 + 3 z^8 + 5 z^7 + z^6 + z^4 + 5 z^3 + 3 z^2 + 3 z + 5$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $24$ |
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | $C_{12}$ (as 12T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $12$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9166666666666666$ |
| Galois splitting model: | $x^{12} - 2 x^{11} - 4 x^{10} - 4 x^{9} - 2 x^{8} + 8 x^{7} + 7 x^{6} + 8 x^{5} - 2 x^{4} - 4 x^{3} - 4 x^{2} - 2 x + 1$ |