Defining polynomial
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\(x^{14} + 2 x^{2} + 2 x + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{8}{7}]$ |
| Visible Swan slopes: | $[\frac{1}{7}]$ |
| Means: | $\langle\frac{1}{14}\rangle$ |
| Rams: | $(1)$ |
| Jump set: | $[7, 15]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.7.6a1.1 |
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: |
\( x^{14} + 2 x^{2} + 2 x + 2 \)
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Ramification polygon
| Residual polynomials: | $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1$,$z + 1$ |
| Associated inertia: | $3$,$1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $168$ |
| Galois group: | $F_8:C_3$ (as 14T11) |
| Inertia group: | $F_8$ (as 14T6) |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $7$ |
| Galois Artin slopes: | $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}]$ |
| Galois Swan slopes: | $[\frac{1}{7},\frac{1}{7},\frac{1}{7}]$ |
| Galois mean slope: | $1.1071428571428572$ |
| Galois splitting model: |
$x^{14} - 7 x^{12} - 28 x^{11} + 7 x^{10} + 168 x^{9} + 147 x^{8} - 272 x^{7} - 413 x^{6} + 364 x^{5} + 511 x^{4} - 196 x^{3} - 427 x^{2} - 420 x + 173$
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