Defining polynomial
|
\(x^{6} - 42 x^{3} + 147\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 7 }) }$: | $6$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{7}(\sqrt{3})$, 7.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified 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: |
\( x^{3} + 7 t \)
$\ \in\Q_{7}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + 3z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | $x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 253 x^{2} + 101 x - 391$ |