Defining polynomial
|
$( x^{4} + 5 x^{2} + 4 x + 3 )^{4} + 7 x$
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{7})$: | $C_8$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $2400 = (7^{ 4 } - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$, 7.4.1.0a1.1, 7.4.2.4a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 7.4.1.0a1.1 $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{4} + 5 x^{2} + 4 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 7 t^{3} \)
$\ \in\Q_{7}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^3 + 4 z^2 + 6 z + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $32$ |
| Galois group: | $\OD_{32}$ (as 16T22) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $8$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.75$ |
| Galois splitting model: | not computed |