Defining polynomial
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\(x^{12} + 2 x^{11} + 2 x^{9} + 6 x^{4} + 2 x^{2} + 2\)
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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 $c$: | $20$ |
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| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ | |
| Root number: | $-i$ | |
| $\Aut(K/\Q_{2})$: | $C_2$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[\frac{4}{3}, \frac{7}{3}]$ | |
| Visible Swan slopes: | $[\frac{1}{3},\frac{4}{3}]$ | |
| Means: | $\langle\frac{1}{6}, \frac{3}{4}\rangle$ | |
| Rams: | $(1, 7)$ | |
| Jump set: | $[3, 7, 19]$ | |
| 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: |
\( x^{12} + 2 x^{11} + 2 x^{9} + 6 x^{4} + 2 x^{2} + 2 \)
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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: | $[9, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $192$ |
| Galois group: | $C_4^2:D_6$ (as 12T115) |
| Inertia group: | $C_4^2:C_6$ (as 12T61) |
| Wild inertia group: | $C_4:D_4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, 2, \frac{7}{3}, \frac{7}{3}]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3}]$ |
| Galois mean slope: | $2.1458333333333335$ |
| Galois splitting model: | $x^{12} + 4 x^{10} - 7 x^{8} + 11 x^{4} + 88 x^{2} + 11$ |