Defining polynomial
|
$( x^{2} + 193 x + 3 )^{8} + 1194 x + 39004$
|
Invariants
| Base field: | $\Q_{199}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{199}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{199})$: | $C_8$ |
| This field is not Galois over $\Q_{199}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $39600 = (199^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{199}(\sqrt{3})$, $\Q_{199}(\sqrt{199})$, $\Q_{199}(\sqrt{199\cdot 3})$, 199.2.2.2a1.2, 199.2.4.6a1.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{199}(\sqrt{3})$ $\cong \Q_{199}(t)$ where $t$ is a root of
\( x^{2} + 193 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{8} + 34228 t + 8358 \)
$\ \in\Q_{199}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 70 z^3 + 56 z^2 + 28 z + 8$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $32$ |
| Galois group: | $C_8.C_4$ (as 16T49) |
| Inertia group: | Intransitive group isomorphic to $C_8$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $8$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.875$ |
| Galois splitting model: | not computed |