\(x^{16} + 8 x^{15} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 6\)
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $56$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$:
|
$C_2$ |
| 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: | $[1, 2, 4, 8, 32]$ |
| Roots of unity: | $2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Galois degree: |
$256$
|
| Galois group: |
$C_2^4.Q_{16}$ (as 16T691)
|
| Inertia group: |
not computed
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$4$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 3, 3, \frac{7}{2}, 4, 4]$
|
| Galois Swan slopes: |
$[1,2,2,\frac{5}{2},3,3]$
|
| Galois mean slope: |
$3.75$
|
| Galois splitting model: | not computed |
