Defining polynomial
|
$( x^{6} + x^{4} + x^{3} + x + 1 )^{3} + 2 x$
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Invariants
| Base field: | $\Q_{2}$ |
|
| Degree $d$: | $18$ |
|
| Ramification index $e$: | $3$ |
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| Residue field degree $f$: | $6$ |
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| Discriminant exponent $c$: | $12$ |
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| Discriminant root field: | $\Q_{2}(\sqrt{5})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$: | $C_9$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | $[3]$ | |
| Roots of unity: | $126 = (2^{ 6 } - 1) \cdot 2$ |
|
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.3.1.0a1.1, 2.6.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.6.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{6} + x^{4} + x^{3} + x + 1 \)
|
|
| Relative Eisenstein polynomial: |
\( x^{3} + 2 t^{2} \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $54$ |
| Galois group: | $S_3\times C_9$ (as 18T16) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $18$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.6666666666666666$ |
| Galois splitting model: | $x^{18} - 8 x^{9} + 64$ |