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 exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[8/7]$ |
Intermediate fields
| 2.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 2 x^{2} + 2 x + 2 \)
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Ramification polygon
| Residual polynomials: | $z + 1$,$z^{12} + z^{10} + z^{8} + z^{6} + z^{4} + z^{2} + 1$ |
| Associated inertia: | $1$,$3$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $F_8:C_3$ (as 14T11) |
| Inertia group: | $F_8$ (as 14T6) |
| Wild inertia group: | $C_2^3$ |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | $[8/7, 8/7, 8/7]$ |
| Galois mean slope: | $31/28$ |
| 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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