\(x^{8} + 8 x^{7} + 4 x^{4} + 10\)
![Copy content]()
![Toggle raw display]()
|
| Base field: | $\Q_{2}$
|
| Degree $d$: | $8$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $30$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{2})$:
|
$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, 4, \frac{19}{4}]$ |
| Visible Swan slopes: | $[2,3,\frac{15}{4}]$ |
| Means: | $\langle1, 2, \frac{23}{8}\rangle$ |
| Rams: | $(2, 4, 7)$ |
| Jump set: | $[1, 3, 7, 15]$ |
| Roots of unity: | $2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Galois degree: |
$64$
|
| Galois group: |
$C_2\wr C_4$ (as 8T27)
|
| Inertia group: |
$C_2\wr C_4$ (as 8T27)
|
| Wild inertia group: |
$C_2\wr C_4$
|
| Galois unramified degree: |
$1$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$
|
| Galois Swan slopes: |
$[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]$
|
| Galois mean slope: |
$4.28125$
|
| Galois splitting model: | $x^{8} + 8 x^{6} - 36 x^{4} - 128 x^{2} - 4$ |
