\(x^{16} + 8 x^{15} + 4 x^{14} + 10 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 16 x^{7} + 8 x^{6} + 2\)
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|
| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $59$ |
| Discriminant root field: | $\Q_{2}(\sqrt{2\cdot 5})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
|
$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[2, 2, \frac{15}{4}, \frac{19}{4}]$ |
| Visible Swan slopes: | $[1,1,\frac{11}{4},\frac{15}{4}]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{4}, \frac{7}{4}, \frac{11}{4}\rangle$ |
| Rams: | $(1, 1, 8, 16)$ |
| Jump set: | $[1, 3, 11, 27, 43]$ |
| Roots of unity: | $4 = 2^{ 2 }$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Galois degree: |
$512$
|
| Galois group: |
$D_8\wr C_2$ (as 16T972)
|
| Inertia group: |
not computed
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$2$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{9}{2}, \frac{19}{4}]$
|
| Galois Swan slopes: |
$[1,1,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{7}{2},\frac{15}{4}]$
|
| Galois mean slope: |
$4.3671875$
|
| Galois splitting model: | not computed |
