Defining polynomial
|
\(x^{6} + 177 x^{3} + 163 x^{2} + 169 x + 2\)
|
Invariants
| Base field: | $\Q_{181}$ |
| Degree $d$: | $6$ |
| Ramification index $e$: | $1$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{181}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{181})$ $=$$\Gal(K/\Q_{181})$: | $C_6$ |
| This field is Galois and abelian over $\Q_{181}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $35161828327080 = (181^{ 6 } - 1)$ |
Intermediate fields
| $\Q_{181}(\sqrt{2})$, 181.3.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 181.6.1.0a1.1 $\cong \Q_{181}(t)$ where $t$ is a root of
\( x^{6} + 177 x^{3} + 163 x^{2} + 169 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x - 181 \)
$\ \in\Q_{181}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois degree: | $6$ |
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.0$ |
| Galois splitting model: | $x^{6} - x^{5} + 2 x^{4} + 8 x^{3} - x^{2} - 5 x + 7$ |