\(x^{8} + 12 x^{6} + 2 x^{4} + 8 x + 2\)
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $8$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$:
|
$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[2, \frac{7}{2}, \frac{15}{4}]$ |
| Visible Swan slopes: | $[1,\frac{5}{2},\frac{11}{4}]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{2}, \frac{17}{8}\rangle$ |
| Rams: | $(1, 4, 5)$ |
| Jump set: | $[1, 5, 13, 21]$ |
| 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: |
$128$
|
| Galois group: |
$C_2\wr D_4$ (as 8T35)
|
| Inertia group: |
$C_2\wr C_2^2$ (as 8T29)
|
| Wild inertia group: |
$C_2\wr C_2^2$
|
| Galois unramified degree: |
$2$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]$
|
| Galois Swan slopes: |
$[1,1,2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]$
|
| Galois mean slope: |
$3.46875$
|
| Galois splitting model: | $x^{8} + 30 x^{4} - 72 x^{2} + 45$ |
