Defining polynomial
|
$( x^{2} + 6 x + 3 )^{9} + 7$
|
Invariants
| Base field: | $\Q_{7}$ |
|
| Degree $d$: | $18$ |
|
| Ramification index $e$: | $9$ |
|
| Residue field degree $f$: | $2$ |
|
| Discriminant exponent $c$: | $16$ |
|
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{7})$: | $C_6$ | |
| This field is not Galois over $\Q_{7}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $48 = (7^{ 2 } - 1)$ |
|
Intermediate fields
| $\Q_{7}(\sqrt{3})$, 7.1.3.2a1.1, 7.2.3.4a1.2, 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: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{2} + 6 x + 3 \)
|
|
| Relative Eisenstein polynomial: |
\( x^{9} + 7 \)
$\ \in\Q_{7}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^8 + 2 z^7 + z^6 + z + 2$ |
| 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: | Intransitive group isomorphic to $C_9$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $9$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.8888888888888888$ |
| Galois splitting model: | not computed |