Defining polynomial
|
\(x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{2} + 6\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_2^2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{4}{3}, 2]$ |
| Visible Swan slopes: | $[\frac{1}{3},1]$ |
| Means: | $\langle\frac{1}{6}, \frac{7}{12}\rangle$ |
| Rams: | $(1, 5)$ |
| Jump set: | $[3, 7, 24]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{-5})$, 2.1.3.2a1.1, 2.1.6.6a1.1, 2.1.6.8a1.3, 2.1.6.8a1.2 |
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: |
\( x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{2} + 6 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $2$,$1$,$1$ |
| Indices of inseparability: | $[7, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $48$ |
| Galois group: | $C_2\times S_4$ (as 12T24) |
| Inertia group: | $C_2\times A_4$ (as 12T7) |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, 2]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},1]$ |
| Galois mean slope: | $1.5833333333333333$ |
| Galois splitting model: | $x^{12} - 2 x^{11} + 8 x^{10} - 4 x^{9} + 5 x^{8} + 10 x^{7} - 14 x^{6} - 10 x^{5} + 5 x^{4} + 4 x^{3} + 8 x^{2} + 2 x + 1$ |