Defining polynomial
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\(x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{4} + 2 x^{2} + 6\)
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $12$ |
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| Ramification index $e$: | $12$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $18$ |
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| Discriminant root field: | $\Q_{2}(\sqrt{5})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{2})$: | $C_2$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[\frac{4}{3}, 2]$ | |
| Visible Swan slopes: | $[\frac{1}{3},1]$ | |
| Means: | $\langle\frac{1}{6}, \frac{7}{12}\rangle$ | |
| Rams: | $(1, 5)$ | |
| Jump set: | $[3, 7, 23]$ | |
| Roots of unity: | $2$ |
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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: | $\Q_{2}$ |
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| Relative Eisenstein polynomial: |
\( x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{4} + 2 x^{2} + 6 \)
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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: | $[7, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $48$ |
| Galois group: | $C_2\times S_4$ (as 12T22) |
| Inertia group: | $C_2\times A_4$ (as 12T6) |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, 2]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},1]$ |
| Galois mean slope: | $1.5833333333333333$ |
| Galois splitting model: | $x^{12} - 4 x^{10} - 2 x^{9} + 17 x^{8} + 6 x^{7} - 30 x^{6} - 6 x^{5} + 31 x^{4} - 26 x^{3} - 4 x^{2} + 28 x - 13$ |