Defining polynomial
|
$( x^{2} + 192 x + 2 )^{9} + 3546 x + 5319$
|
Invariants
| Base field: | $\Q_{197}$ |
| Degree $d$: | $18$ |
| Ramification index $e$: | $9$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{197}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{197})$: | $C_9$ |
| This field is not Galois over $\Q_{197}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $38808 = (197^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{197}(\sqrt{2})$, 197.1.3.2a1.1 x3, 197.2.3.4a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{197}(\sqrt{2})$ $\cong \Q_{197}(t)$ where $t$ is a root of
\( x^{2} + 192 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{9} + 3546 t + 5319 \)
$\ \in\Q_{197}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^8 + 9 z^7 + 36 z^6 + 84 z^5 + 126 z^4 + 126 z^3 + 84 z^2 + 36 z + 9$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $54$ |
| Galois group: | $C_3\times D_9$ (as 18T19) |
| 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 |