Defining polynomial
|
\(x^{12} + 8 x^{11} + 8 x^{7} + 8 x^{5} + 8 x + 10\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $35$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[3, 4]$ |
| Visible Swan slopes: | $[2,3]$ |
| Means: | $\langle1, 2\rangle$ |
| Rams: | $(6, 12)$ |
| Jump set: | $[3, 9, 21]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{-2\cdot 5})$, 2.1.3.2a1.1, 2.1.6.11a1.2 |
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: |
\( x^{12} + 8 x^{11} + 8 x^{7} + 8 x^{5} + 8 x + 10 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $2$,$1$,$1$ |
| Indices of inseparability: | $[24, 12, 0]$ |
Invariants of the Galois closure
| Galois degree: | $384$ |
| Galois group: | $C_2^4:D_{12}$ (as 12T154) |
| Inertia group: | $C_2^4:C_{12}$ (as 12T105) |
| Wild inertia group: | $C_2^4:C_4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]$ |
| Galois Swan slopes: | $[\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]$ |
| Galois mean slope: | $3.7604166666666665$ |
| Galois splitting model: | $x^{12} + 4 x^{10} + 70 x^{8} + 88 x^{6} - 682 x^{4} + 484 x^{2} - 242$ |