Defining polynomial
|
\(x^{18} + 7\)
|
Invariants
| Base field: | $\Q_{7}$ |
|
| Degree $d$: | $18$ |
|
| Ramification index $e$: | $18$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $17$ |
|
| 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: | $[3]$ | |
| Roots of unity: | $42 = (7 - 1) \cdot 7$ |
|
Intermediate fields
| $\Q_{7}(\sqrt{7\cdot 3})$, 7.1.3.2a1.1, 7.1.6.5a1.1, 7.1.9.8a1.1 |
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: |
\( x^{18} + 7 \)
|
Ramification polygon
| Residual polynomials: | $z^{17} + 4 z^{16} + 6 z^{15} + 4 z^{14} + z^{13} + 2 z^{10} + z^9 + 5 z^8 + z^7 + 2 z^6 + z^3 + 4 z^2 + 6 z + 4$ |
| Associated inertia: | $3$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $54$ |
| Galois group: | $C_9:C_6$ (as 18T14) |
| Inertia group: | $C_{18}$ (as 18T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $18$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9444444444444444$ |
| Galois splitting model: | not computed |