\(x^{16} + 8 x^{12} + 16 x^{10} + 4 x^{8} + 16 x^{5} + 10\)
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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 $c$: | $68$ |
| 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: | $[3, 4, \frac{19}{4}, \frac{19}{4}]$ |
| Visible Swan slopes: | $[2,3,\frac{15}{4},\frac{15}{4}]$ |
| Means: | $\langle1, 2, \frac{23}{8}, \frac{53}{16}\rangle$ |
| Rams: | $(2, 4, 7, 7)$ |
| 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.
| Galois degree: |
$2048$
|
| Galois group: |
$C_2\wr (C_2\times C_4)$ (as 16T1385)
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| Inertia group: |
$C_2^2\wr C_4$ (as 16T1224)
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| Wild inertia group: |
not computed
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| Galois unramified degree: |
$2$
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| Galois tame degree: |
$1$
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| Galois Artin slopes: |
$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}, \frac{19}{4}, \frac{19}{4}]$
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| Galois Swan slopes: |
$[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4},\frac{15}{4},\frac{15}{4}]$
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| Galois mean slope: |
$4.583984375$
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| Galois splitting model: |
$x^{16} + 16 x^{14} + 232 x^{12} + 1648 x^{10} - 1520 x^{9} + 8100 x^{8} - 11520 x^{7} + 31072 x^{6} - 39760 x^{5} + 75792 x^{4} - 68800 x^{3} + 67584 x^{2} + 4160 x + 15786$
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