Defining polynomial
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\(x^{16} + 4 x^{14} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x^{3} + 8 x + 6\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $48$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_2^2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 3, 3, \frac{7}{2}]$ |
| Visible Swan slopes: | $[1,2,2,\frac{5}{2}]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{13}{8}, \frac{33}{16}\rangle$ |
| Rams: | $(1, 3, 3, 7)$ |
| Jump set: | $[1, 2, 4, 8, 32]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{-5})$, 2.1.4.8b1.4, 2.1.4.9a1.8, 2.1.4.9a1.5, 2.1.8.20d1.12, 2.1.8.22d1.27, 2.1.8.22d1.29 |
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: |
\( x^{16} + 4 x^{14} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x^{3} + 8 x + 6 \)
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Ramification polygon
| Residual polynomials: | $z^8 + 1$,$z^6 + 1$,$z + 1$ |
| Associated inertia: | $1$,$2$,$1$ |
| Indices of inseparability: | $[33, 26, 24, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $128$ |
| Galois group: | $C_2^4.D_4$ (as 16T230) |
| Inertia group: | $C_2^3:C_4$ (as 16T21) |
| Wild inertia group: | $C_2^3:C_4$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 3, 3, \frac{7}{2}]$ |
| Galois Swan slopes: | $[1,1,2,2,\frac{5}{2}]$ |
| Galois mean slope: | $3.0625$ |
| Galois splitting model: |
$x^{16} + 8 x^{14} + 64 x^{12} + 312 x^{10} + 746 x^{8} - 1080 x^{6} + 640 x^{4} - 200 x^{2} + 25$
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