Defining polynomial
|
$( x^{2} + 6 x + 3 )^{3} + 7$
|
Invariants
| Base field: | $\Q_{7}$ |
|
| Degree $d$: | $6$ |
|
| Ramification index $e$: | $3$ |
|
| Residue field degree $f$: | $2$ |
|
| Discriminant exponent $c$: | $4$ |
|
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{7})$ $=$ $\Gal(K/\Q_{7})$: | $C_6$ | |
| This field is Galois and abelian 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 |
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: |
\( x^{3} + 7 \)
$\ \in\Q_{7}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^2 + 3 z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $6$ |
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.6666666666666666$ |
| Galois splitting model: | $x^{6} + 5 x^{4} + 6 x^{2} + 1$ |