Defining polynomial
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\(x^{12} + 4 x^{11} + 4 x^{9} + 2 x^{4} + 2 x^{2} + 4 x + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{4}{3}, 3]$ |
| Visible Swan slopes: | $[\frac{1}{3},2]$ |
| Means: | $\langle\frac{1}{6}, \frac{13}{12}\rangle$ |
| Rams: | $(1, 11)$ |
| Jump set: | $[3, 7, 19]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.3.2a1.1, 2.1.6.6a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 4 x^{11} + 4 x^{9} + 2 x^{4} + 2 x^{2} + 4 x + 2 \)
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Ramification polygon
| Residual polynomials: | $z^8 + z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $2$,$1$,$1$ |
| Indices of inseparability: | $[13, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $384$ |
| Galois group: | $C_2^4:S_4$ (as 12T137) |
| Inertia group: | $C_2^4:A_4$ (as 12T88) |
| Wild inertia group: | $C_2^3:D_4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2]$ |
| Galois mean slope: | $2.6979166666666665$ |
| Galois splitting model: |
$x^{12} - 8 x^{10} + 47 x^{8} - 184 x^{6} + 49 x^{4} + 686 x^{2} + 343$
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