Defining polynomial
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\(x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 10 x^{8} + 16 x^{7} + 4 x^{4} + 16 x^{3} + 16 x + 14\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $58$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $Q_8$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 3, \frac{7}{2}, \frac{9}{2}]$ |
| Visible Swan slopes: | $[1,2,\frac{5}{2},\frac{7}{2}]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{15}{8}, \frac{43}{16}\rangle$ |
| Rams: | $(1, 3, 5, 13)$ |
| Jump set: | $[1, 2, 4, 8, 32]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, $\Q_{2}(\sqrt{2})$, 2.1.4.10a1.7 x2, 2.1.4.9a1.6 x2, 2.1.4.8b1.6, 2.1.8.22d1.23 |
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: |
\( x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 10 x^{8} + 16 x^{7} + 4 x^{4} + 16 x^{3} + 16 x + 14 \)
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Ramification polygon
| Residual polynomials: | $z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$,$1$,$1$ |
| Indices of inseparability: | $[43, 30, 20, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $32$ |
| Galois group: | $Q_{16}:C_2$ (as 16T50) |
| Inertia group: | $\SD_{16}$ (as 16T12) |
| Wild inertia group: | $\SD_{16}$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 3, \frac{7}{2}, \frac{9}{2}]$ |
| Galois Swan slopes: | $[1,2,\frac{5}{2},\frac{7}{2}]$ |
| Galois mean slope: | $3.625$ |
| Galois splitting model: | $x^{16} - 84 x^{12} + 1770 x^{8} - 1404 x^{4} + 9$ |