\(x^{16} + 2 x^{15} + 4 x^{12} + 4 x^{3} + 2 x^{2} + 2\)
![Copy content]()
![Toggle raw display]()
|
| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $30$ |
| 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: | $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{11}{4}]$ |
| Visible Swan slopes: | $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{7}{4}]$ |
| Means: | $\langle\frac{1}{14}, \frac{3}{28}, \frac{1}{8}, \frac{15}{16}\rangle$ |
| Rams: | $(\frac{1}{7}, \frac{1}{7}, \frac{1}{7}, 13)$ |
| Jump set: | $[1, 2, 4, 9, 25]$ |
| Roots of unity: | $2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Galois degree: |
$21504$
|
| Galois group: |
$C_2^7:F_8:C_3$ (as 16T1800)
|
| Inertia group: |
$C_2^7:F_8$ (as 16T1694)
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$3$
|
| Galois tame degree: |
$7$
|
| Galois Artin slopes: |
$[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{18}{7}, \frac{18}{7}, \frac{18}{7}, \frac{19}{7}, \frac{19}{7}, \frac{19}{7}, \frac{11}{4}]$
|
| Galois Swan slopes: |
$[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{11}{7},\frac{11}{7},\frac{11}{7},\frac{12}{7},\frac{12}{7},\frac{12}{7},\frac{7}{4}]$
|
| Galois mean slope: |
$2.7117745535714284$
|
| Galois splitting model: |
$x^{16} - 4 x^{14} + 28 x^{12} + 112 x^{8} + 560 x^{6} + 672 x^{4} + 320 x^{2} + 64$
![Copy content]()
![Toggle raw display]()
|
