Defining polynomial
|
\(x^{6} + 2 x^{5} + 4 x^{4} + 4 x^{3} + 4 x + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
|
| Degree $d$: | $6$ |
|
| Ramification index $e$: | $6$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $10$ |
|
| 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: | $[\frac{8}{3}]$ | |
| Visible Swan slopes: | $[\frac{5}{3}]$ | |
| Means: | $\langle\frac{5}{6}\rangle$ | |
| Rams: | $(5)$ | |
| Jump set: | $[3, 9]$ | |
| Roots of unity: | $2$ |
|
Intermediate fields
| 2.1.3.2a1.1 |
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: |
\( x^{6} + 2 x^{5} + 4 x^{4} + 4 x^{3} + 4 x + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^4 + z^2 + 1$,$z + 1$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
| Galois degree: | $24$ |
| Galois group: | $S_4$ (as 6T7) |
| Inertia group: | $A_4$ (as 6T4) |
| Wild inertia group: | $C_2^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{8}{3}, \frac{8}{3}]$ |
| Galois Swan slopes: | $[\frac{5}{3},\frac{5}{3}]$ |
| Galois mean slope: | $2.1666666666666665$ |
| Galois splitting model: | $x^{6} - 2 x^{4} + 8 x^{2} - 4$ |