Defining polynomial
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$( x^{2} + 108 x + 6 )^{3} + 109 x$
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Invariants
| Base field: | $\Q_{109}$ |
| Degree $d$: | $6$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{109}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{109})$ $=$$\Gal(K/\Q_{109})$: | $C_6$ |
| This field is Galois and abelian over $\Q_{109}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $11880 = (109^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{109}(\sqrt{2})$, 109.1.3.2a1.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{109}(\sqrt{2})$ $\cong \Q_{109}(t)$ where $t$ is a root of
\( x^{2} + 108 x + 6 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 109 t \)
$\ \in\Q_{109}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + 3 z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $6$ |
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.6666666666666666$ |
| Galois splitting model: |
$x^{6} - x^{5} + 2 x^{4} + 5138 x^{3} + 103321 x^{2} + 767139 x + 5027445$
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