\(x^{16} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 4 x^{4} + 6\)
![Copy content]()
![Toggle raw display]()
|
| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $58$ |
| 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: | $[2, 3, \frac{25}{6}, \frac{25}{6}]$ |
| Visible Swan slopes: | $[1,2,\frac{19}{6},\frac{19}{6}]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{53}{24}, \frac{43}{16}\rangle$ |
| Rams: | $(1, 3, \frac{23}{3}, \frac{23}{3})$ |
| 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: |
$6144$
|
| Galois group: |
$C_2^6.(C_2^2\times S_4)$ (as 16T1668)
|
| Inertia group: |
$C_2^8:(C_2\times C_6)$ (as 16T1516)
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$2$
|
| Galois tame degree: |
$3$
|
| Galois Artin slopes: |
$[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{10}{3}, \frac{10}{3}, \frac{23}{6}, \frac{23}{6}, \frac{25}{6}, \frac{25}{6}]$
|
| Galois Swan slopes: |
$[1,\frac{5}{3},\frac{5}{3},2,\frac{7}{3},\frac{7}{3},\frac{17}{6},\frac{17}{6},\frac{19}{6},\frac{19}{6}]$
|
| Galois mean slope: |
$4.041666666666667$
|
| Galois splitting model: |
$x^{16} - 8 x^{14} - 32 x^{13} + 240 x^{12} - 680 x^{11} + 1536 x^{10} - 1904 x^{9} - 802 x^{8} + 3248 x^{7} + 2816 x^{6} - 6544 x^{5} - 3692 x^{4} + 4976 x^{3} + 2592 x^{2} - 176 x + 22$
![Copy content]()
![Toggle raw display]()
|
