\(x^{16} + 16 x^{15} + 4 x^{12} + 16 x^{11} + 8 x^{6} + 16 x^{3} + 34\)
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| Base field: | $\Q_{2}$
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| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $66$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
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$C_4$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, \frac{7}{2}, 4, 5]$ |
| Visible Swan slopes: | $[2,\frac{5}{2},3,4]$ |
| Means: | $\langle1, \frac{7}{4}, \frac{19}{8}, \frac{51}{16}\rangle$ |
| Rams: | $(2, 3, 5, 13)$ |
| Jump set: | $[1, 3, 7, 15, 31]$ |
| Roots of unity: | $2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Residual polynomials: | $z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$,$1$,$1$ |
| Indices of inseparability: | $[51, 38, 28, 16, 0]$ |
| Galois degree: |
$512$
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| Galois group: |
$C_4^3:D_4$ (as 16T890)
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| Inertia group: |
$D_4^2:C_4$ (as 16T503)
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| Wild inertia group: |
$D_4^2:C_4$
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| Galois unramified degree: |
$2$
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| Galois tame degree: |
$1$
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| Galois Artin slopes: |
$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, 4, 5]$
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| Galois Swan slopes: |
$[1,1,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},3,4]$
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| Galois mean slope: |
$4.3671875$
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| Galois splitting model: |
$x^{16} - 24 x^{14} + 260 x^{12} - 1512 x^{10} + 4546 x^{8} - 6360 x^{6} + 4060 x^{4} - 1000 x^{2} + 25$
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