Defining polynomial
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\(x^{16} + 4 x^{14} + 2 x^{12} + 8 x^{8} + 8 x^{6} + 8 x^{2} + 8 x + 2\)
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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: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 2, \frac{7}{2}, \frac{7}{2}]$ |
| Visible Swan slopes: | $[1,1,\frac{5}{2},\frac{5}{2}]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{4}, \frac{13}{8}, \frac{33}{16}\rangle$ |
| Rams: | $(1, 1, 7, 7)$ |
| Jump set: | $[1, 3, 11, 27, 43]$ |
| Roots of unity: | $4 = 2^{ 2 }$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.1.4.6a1.1, 2.1.8.20b1.3 |
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: |
\( x^{16} + 4 x^{14} + 2 x^{12} + 8 x^{8} + 8 x^{6} + 8 x^{2} + 8 x + 2 \)
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Ramification polygon
| Residual polynomials: | $z^{12} + 1$,$z^3 + 1$ |
| Associated inertia: | $2$,$2$ |
| Indices of inseparability: | $[33, 28, 12, 12, 0]$ |
Invariants of the Galois closure
| Galois degree: | $128$ |
| Galois group: | $C_2^4.D_4$ (as 16T297) |
| Inertia group: | $C_4^2:C_2$ (as 16T30) |
| Wild inertia group: | $C_4^2:C_2$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]$ |
| Galois Swan slopes: | $[1,1,2,\frac{5}{2},\frac{5}{2}]$ |
| Galois mean slope: | $3.1875$ |
| Galois splitting model: |
$x^{16} - 8 x^{15} + 32 x^{14} - 80 x^{13} + 130 x^{12} - 208 x^{11} + 344 x^{10} - 496 x^{9} + 1240 x^{8} - 2760 x^{7} + 5376 x^{6} - 9288 x^{5} + 12672 x^{4} - 11880 x^{3} + 7560 x^{2} - 3240 x + 810$
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