Defining polynomial
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\(x^{16} + 16 x^{15} + 16 x^{13} + 8 x^{10} + 2 x^{8} + 16 x^{7} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 10\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $64$ |
| 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: | $[2, 3, 4, 5]$ |
| Visible Swan slopes: | $[1,2,3,4]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{17}{8}, \frac{49}{16}\rangle$ |
| Rams: | $(1, 3, 7, 15)$ |
| Jump set: | $[1, 2, 4, 32, 48]$ |
| Roots of unity: | $4 = 2^{ 2 }$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2\cdot 5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.1.4.8b1.2, 2.1.4.11a1.6 x2, 2.1.4.11a1.19 x2, 2.1.8.24c1.18 |
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: |
\( x^{16} + 16 x^{15} + 16 x^{13} + 8 x^{10} + 2 x^{8} + 16 x^{7} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 10 \)
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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: | $[49, 34, 20, 8, 0]$ |
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: | $[2, 3, 4, 5]$ |
| Galois Swan slopes: | $[1,2,3,4]$ |
| Galois mean slope: | $4.0$ |
| Galois splitting model: |
$x^{16} + 140 x^{8} + 62500$
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