Defining polynomial
|
\(x^{6} + 7 x^{4} + 91 x^{3} + 55 x^{2} + 109 x + 2\)
|
Invariants
| Base field: | $\Q_{179}$ |
|
| Degree $d$: | $6$ |
|
| Ramification index $e$: | $1$ |
|
| Residue field degree $f$: | $6$ |
|
| Discriminant exponent $c$: | $0$ |
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| Discriminant root field: | $\Q_{179}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{179})$ $=$ $\Gal(K/\Q_{179})$: | $C_6$ | |
| This field is Galois and abelian over $\Q_{179}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $32894113444920 = (179^{ 6 } - 1)$ |
|
Intermediate fields
| $\Q_{179}(\sqrt{2})$, 179.3.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 179.6.1.0a1.1 $\cong \Q_{179}(t)$ where $t$ is a root of
\( x^{6} + 7 x^{4} + 91 x^{3} + 55 x^{2} + 109 x + 2 \)
|
|
| Relative Eisenstein polynomial: |
\( x - 179 \)
$\ \in\Q_{179}(t)[x]$
|
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: | $x^{6} - x^{5} + 3 x^{4} - 11 x^{3} + 44 x^{2} - 36 x + 32$ |