\(x^{16} + 8 x^{15} + 8 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 2\)
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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 $c$: | $56$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$:
|
$C_1$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[2, 3, 4, 4]$ |
| Visible Swan slopes: | $[1,2,3,3]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{17}{8}, \frac{41}{16}\rangle$ |
| Rams: | $(1, 3, 7, 7)$ |
| Jump set: | $[1, 11, 25, 41, 57]$ |
| Roots of unity: | $8 = 2^{ 3 }$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Galois degree: |
$3072$
|
| Galois group: |
$C_2^8:(C_2\times C_6)$ (as 16T1516)
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| Inertia group: |
$C_2^2\wr C_2^2$ (as 16T1152)
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| Wild inertia group: |
not computed
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| Galois unramified degree: |
$3$
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| Galois tame degree: |
$1$
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| Galois Artin slopes: |
$[2, 2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, 4, 4]$
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| Galois Swan slopes: |
$[1,1,1,2,2,2,\frac{5}{2},\frac{5}{2},3,3]$
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| Galois mean slope: |
$3.833984375$
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| Galois splitting model: |
$x^{16} + 24 x^{14} - 32 x^{13} + 232 x^{12} - 464 x^{11} + 1432 x^{10} - 2616 x^{9} + 4496 x^{8} - 5728 x^{7} + 4896 x^{6} - 1056 x^{5} - 2144 x^{4} + 816 x^{3} + 10368 x^{2} - 25912 x + 16361$
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