Defining polynomial
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\(x^{14} + 2 x^{10} + 2 x^{5} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $14$ |
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| Ramification index $e$: | $14$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $18$ |
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| Discriminant root field: | $\Q_{2}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$: | $C_2$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[\frac{12}{7}]$ | |
| Visible Swan slopes: | $[\frac{5}{7}]$ | |
| Means: | $\langle\frac{5}{14}\rangle$ | |
| Rams: | $(5)$ | |
| Jump set: | $[7, 19]$ | |
| Roots of unity: | $2$ |
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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: | $\Q_{2}$ |
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| Relative Eisenstein polynomial: |
\( x^{14} + 2 x^{10} + 2 x^{5} + 2 \)
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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: | $[5, 0]$ |
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: | $[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}]$ |
| Galois Swan slopes: | $[\frac{5}{7},\frac{5}{7},\frac{5}{7}]$ |
| Galois mean slope: | $1.6071428571428572$ |
| Galois splitting model: |
$x^{14} + 7 x^{12} + 35 x^{10} - 14 x^{9} - 203 x^{8} - 356 x^{7} + 105 x^{6} + 518 x^{5} + 539 x^{4} - 532 x^{3} - 287 x^{2} + 196 x - 23$
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