\(x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 2 x^{10} + 4 x^{7} + 2 x^{4} + 8 x^{3} + 2\)
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $38$ |
| 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: | $[\frac{4}{3}, \frac{4}{3}, 2, \frac{13}{4}]$ |
| Visible Swan slopes: | $[\frac{1}{3},\frac{1}{3},1,\frac{9}{4}]$ |
| Means: | $\langle\frac{1}{6}, \frac{1}{4}, \frac{5}{8}, \frac{23}{16}\rangle$ |
| Rams: | $(\frac{1}{3}, \frac{1}{3}, 3, 13)$ |
| Jump set: | $[1, 2, 5, 21, 37]$ |
| 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: |
$768$
|
| Galois group: |
$C_2^5:S_4$ (as 16T1044)
|
| Inertia group: |
not computed
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$2$
|
| Galois tame degree: |
$3$
|
| Galois Artin slopes: |
$[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, \frac{13}{4}]$
|
| Galois Swan slopes: |
$[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},\frac{9}{4}]$
|
| Galois mean slope: |
$3.0989583333333335$
|
| Galois splitting model: | not computed |
