Defining polynomial
|
\(x^{12} + 4 x^{10} + 4 x^{8} + 8 x^{6} + 12 x^{4} + 8 x^{3} + 26\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $35$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[3, 4]$ |
Intermediate fields
| 2.3.2.1, 2.6.11.14 |
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: |
\( x^{12} + 4 x^{10} + 4 x^{8} + 8 x^{6} + 12 x^{4} + 8 x^{3} + 26 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{8} + z^{4} + 1$ |
| Associated inertia: | $1$,$1$,$2$ |
| Indices of inseparability: | $[24, 12, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^4:D_{12}$ (as 12T152) |
| Inertia group: | $C_2^4:C_{12}$ (as 12T99) |
| Wild inertia group: | $C_2^4:C_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[4/3, 4/3, 3, 19/6, 19/6, 4]$ |
| Galois mean slope: | $331/96$ |
| Galois splitting model: |
$x^{12} + 12 x^{10} - 22 x^{8} + 80 x^{6} - 100 x^{4} + 48 x^{2} - 8$
|