\(x^{16} + 8 x^{14} + 16 x^{13} + 16 x^{12} + 16 x^{10} + 4 x^{8} + 16 x^{7} + 18\)
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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 $c$: | $70$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$:
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$C_2^2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, 4, \frac{19}{4}, 5]$ |
| Visible Swan slopes: | $[2,3,\frac{15}{4},4]$ |
| Means: | $\langle1, 2, \frac{23}{8}, \frac{55}{16}\rangle$ |
| Rams: | $(2, 4, 7, 9)$ |
| 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: | $[55, 46, 32, 16, 0]$ |
| Galois degree: |
$128$
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| Galois group: |
$C_2^4:C_8$ (as 16T228)
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| Inertia group: |
$C_2^4:C_8$ (as 16T228)
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| Wild inertia group: |
$C_2^4:C_8$
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| Galois unramified degree: |
$1$
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| Galois tame degree: |
$1$
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| Galois Artin slopes: |
$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$
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| Galois Swan slopes: |
$[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$
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| Galois mean slope: |
$4.640625$
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| Galois splitting model: |
$x^{16} - 16 x^{14} - 16 x^{13} + 92 x^{12} + 352 x^{11} - 352 x^{10} + 1200 x^{9} + 2082 x^{8} - 29936 x^{7} - 18528 x^{6} + 144544 x^{5} + 64836 x^{4} - 417072 x^{3} + 277392 x^{2} + 4576 x - 31409$
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