\(x^{16} + 8 x^{14} + 16 x^{13} + 18 x^{8} + 16 x^{7} + 16 x^{3} + 2\)
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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 $c$: | $66$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
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$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[2, \frac{7}{2}, \frac{9}{2}, \frac{39}{8}]$ |
| Visible Swan slopes: | $[1,\frac{5}{2},\frac{7}{2},\frac{31}{8}]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \frac{51}{16}\rangle$ |
| Rams: | $(1, 4, 8, 11)$ |
| 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: | $[51, 40, 24, 8, 0]$ |
| Galois degree: |
$256$
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| Galois group: |
$C_2^4.D_8$ (as 16T705)
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| Inertia group: |
$C_2^4.D_8$ (as 16T705)
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| Wild inertia group: |
$C_2^4.D_8$
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| Galois unramified degree: |
$1$
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| Galois tame degree: |
$1$
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| Galois Artin slopes: |
$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}]$
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| Galois Swan slopes: |
$[1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8}]$
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| Galois mean slope: |
$4.6640625$
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| Galois splitting model: |
$x^{16} - 32 x^{14} - 16 x^{13} + 456 x^{12} + 128 x^{11} - 3184 x^{10} + 224 x^{9} + 13116 x^{8} - 6432 x^{7} - 26600 x^{6} + 32096 x^{5} + 7548 x^{4} - 42736 x^{3} + 39976 x^{2} - 18112 x + 3769$
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