Defining polynomial
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\(x^{6} + 4 x^{3} + 4 x + 10\)
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $6$ |
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| Ramification index $e$: | $6$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $11$ |
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| Discriminant root field: | $\Q_{2}(\sqrt{-2\cdot 5})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{2})$: | $C_2$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[3]$ | |
| Visible Swan slopes: | $[2]$ | |
| Means: | $\langle1\rangle$ | |
| Rams: | $(6)$ | |
| Jump set: | $[3, 9]$ | |
| Roots of unity: | $2$ |
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Intermediate fields
| 2.1.3.2a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
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| Relative Eisenstein polynomial: |
\( x^{6} + 4 x^{3} + 4 x + 10 \)
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Ramification polygon
| Residual polynomials: | $z^4 + z^2 + 1$,$z + 1$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $48$ |
| Galois group: | $C_2\times S_4$ (as 6T11) |
| Inertia group: | $C_2\times A_4$ (as 6T6) |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{8}{3}, \frac{8}{3}, 3]$ |
| Galois Swan slopes: | $[\frac{5}{3},\frac{5}{3},2]$ |
| Galois mean slope: | $2.5833333333333335$ |
| Galois splitting model: |
$x^{6} + 2 x^{4} + 2 x^{2} + 10$
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