Defining polynomial
|
\(x^{12} + 4 x^{11} + 4 x^{5} + 2 x^{4} + 10 x^{2} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $25$ |
| 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: | $[\frac{4}{3}, \frac{19}{6}]$ |
| Visible Swan slopes: | $[\frac{1}{3},\frac{13}{6}]$ |
| Means: | $\langle\frac{1}{6}, \frac{7}{6}\rangle$ |
| Rams: | $(1, 12)$ |
| Jump set: | $[3, 7, 19]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.3.2a1.1, 2.1.6.6a1.2 |
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: |
\( x^{12} + 4 x^{11} + 4 x^{5} + 2 x^{4} + 10 x^{2} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $2$,$1$,$1$ |
| Indices of inseparability: | $[14, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $384$ |
| Galois group: | $C_2^4:S_4$ (as 12T137) |
| Inertia group: | $C_2^4:A_4$ (as 12T88) |
| Wild inertia group: | $C_2^3:D_4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6}]$ |
| Galois mean slope: | $2.9479166666666665$ |
| Galois splitting model: | $x^{12} - 4 x^{10} + 6 x^{8} - 4 x^{6} - 76 x^{4} + 88$ |