\(x^{16} + 16 x^{11} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 16 x^{5} + 16 x^{4} + 16 x^{3} + 16 x + 2\)
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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 $c$: | $64$ |
| 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: | $[2, \frac{7}{2}, \frac{17}{4}, \frac{19}{4}]$ |
| Visible Swan slopes: | $[1,\frac{5}{2},\frac{13}{4},\frac{15}{4}]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{2}, \frac{19}{8}, \frac{49}{16}\rangle$ |
| Rams: | $(1, 4, 7, 11)$ |
| Jump set: | $[1, 5, 13, 29, 45]$ |
| Roots of unity: | $4 = 2^{ 2 }$ |
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: | $[49, 38, 24, 8, 0]$ |
| Galois degree: |
$128$
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| Galois group: |
$C_2^4.D_4$ (as 16T297)
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| Inertia group: |
$C_4^2:C_4$ (as 16T143)
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| Wild inertia group: |
$C_4^2:C_4$
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| Galois unramified degree: |
$2$
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| Galois tame degree: |
$1$
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| Galois Artin slopes: |
$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$
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| Galois Swan slopes: |
$[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]$
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| Galois mean slope: |
$4.28125$
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| Galois splitting model: |
$x^{16} - 8 x^{14} + 68 x^{12} - 208 x^{10} + 568 x^{8} - 592 x^{6} + 680 x^{4} - 320 x^{2} + 100$
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