Defining polynomial
|
\(x^{8} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $18$ |
| 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: | $[2, \frac{8}{3}, \frac{8}{3}]$ |
| Visible Swan slopes: | $[1,\frac{5}{3},\frac{5}{3}]$ |
| Means: | $\langle\frac{1}{2}, \frac{13}{12}, \frac{11}{8}\rangle$ |
| Rams: | $(1, \frac{7}{3}, \frac{7}{3})$ |
| Jump set: | $[1, 2, 11, 19]$ |
| Roots of unity: | $4 = 2^{ 2 }$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$ |
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: |
\( x^{8} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^4 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[11, 10, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $192$ |
| Galois group: | $C_2^3:S_4$ (as 8T41) |
| Inertia group: | $C_2^3:A_4$ (as 8T33) |
| Wild inertia group: | $C_2^2\wr C_2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]$ |
| Galois mean slope: | $2.3958333333333335$ |
| Galois splitting model: | $x^{8} - 4 x^{7} + 16 x^{6} - 32 x^{5} + 56 x^{4} - 56 x^{3} + 44 x^{2} - 20 x + 5$ |