\(x^{16} + 2 x^{11} + 2 x^{9} + 2 x^{6} + 2 x^{4} + 2\)
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$:
|
$C_4$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, \frac{3}{2}, \frac{7}{4}]$ |
| Visible Swan slopes: | $[\frac{1}{3},\frac{1}{3},\frac{1}{2},\frac{3}{4}]$ |
| Means: | $\langle\frac{1}{6}, \frac{1}{4}, \frac{3}{8}, \frac{9}{16}\rangle$ |
| Rams: | $(\frac{1}{3}, \frac{1}{3}, 1, 3)$ |
| Jump set: | $[1, 2, 5, 11, 25]$ |
| Roots of unity: | $2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Galois degree: |
$768$
|
| Galois group: |
$C_2^4.\GL(2,3)$ (as 16T1064)
|
| Inertia group: |
not computed
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| Wild inertia group: |
not computed
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| Galois unramified degree: |
$4$
|
| Galois tame degree: |
$3$
|
| Galois Artin slopes: |
$[\frac{4}{3}, \frac{4}{3}, \frac{3}{2}, \frac{5}{3}, \frac{5}{3}, \frac{7}{4}]$
|
| Galois Swan slopes: |
$[\frac{1}{3},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{2}{3},\frac{3}{4}]$
|
| Galois mean slope: |
$1.6666666666666667$
|
| Galois splitting model: | not computed |
