Defining polynomial
|
\(x^{12} + 2 x^{10} + 4 x^{9} + 8 x^{8} + 12 x^{6} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 6\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $32$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[8/3, 11/3]$ |
Intermediate fields
| 2.3.2.1, 2.6.10.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 2 x^{10} + 4 x^{9} + 8 x^{8} + 12 x^{6} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 6 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{8} + z^{4} + 1$ |
| Associated inertia: | $1$,$1$,$2$ |
| Indices of inseparability: | $[21, 10, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^4.S_4$ (as 12T140) |
| Inertia group: | $C_2^4.A_4$ (as 12T92) |
| Wild inertia group: | $C_4^2:C_2^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[2, 8/3, 8/3, 3, 11/3, 11/3]$ |
| Galois mean slope: | $41/12$ |
| Galois splitting model: | $x^{12} + 6 x^{8} + 12 x^{4} - 4$ |