Defining polynomial
|
\(x^{8} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 16 x + 6\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $27$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2\cdot 5})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, \frac{7}{2}, \frac{9}{2}]$ |
| Visible Swan slopes: | $[1,\frac{5}{2},\frac{7}{2}]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{2}, \frac{5}{2}\rangle$ |
| Rams: | $(1, 4, 8)$ |
| Jump set: | $[1, 2, 4, 16]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{-5})$, 2.1.4.9a1.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{8} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 16 x + 6 \)
|
Ramification polygon
| Residual polynomials: | $z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$,$1$ |
| Indices of inseparability: | $[20, 12, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $64$ |
| Galois group: | $D_4:D_4$ (as 8T26) |
| Inertia group: | $C_4\wr C_2$ (as 8T17) |
| Wild inertia group: | $C_4\wr C_2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 3, \frac{7}{2}, 4, \frac{9}{2}]$ |
| Galois Swan slopes: | $[1,2,\frac{5}{2},3,\frac{7}{2}]$ |
| Galois mean slope: | $3.9375$ |
| Galois splitting model: | $x^{8} + 6 x^{4} + 6$ |