\(x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 4 x^{8} + 8 x^{6} + 24 x^{4} + 16 x + 2\)
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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 $c$: | $60$ |
| 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: | $[3, \frac{7}{2}, 4, \frac{17}{4}]$ |
| Visible Swan slopes: | $[2,\frac{5}{2},3,\frac{13}{4}]$ |
| Means: | $\langle1, \frac{7}{4}, \frac{19}{8}, \frac{45}{16}\rangle$ |
| Rams: | $(2, 3, 5, 7)$ |
| 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: | $[45, 38, 28, 16, 0]$ |
| Galois degree: |
$128$
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| Galois group: |
$C_2^4:D_4$ (as 16T392)
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| Inertia group: |
$C_2^4:C_4$ (as 16T93)
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| Wild inertia group: |
$C_2^4:C_4$
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| Galois unramified degree: |
$2$
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| Galois tame degree: |
$1$
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| Galois Artin slopes: |
$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]$
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| Galois Swan slopes: |
$[1,1,2,\frac{5}{2},3,\frac{13}{4}]$
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| Galois mean slope: |
$3.84375$
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| Galois splitting model: |
$x^{16} - 8 x^{14} - 8 x^{13} + 136 x^{12} + 368 x^{11} + 184 x^{10} - 728 x^{9} - 1098 x^{8} - 896 x^{7} - 1712 x^{6} + 3752 x^{5} + 6384 x^{4} - 5552 x^{3} + 2096 x^{2} - 456 x + 51$
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