Defining polynomial
|
\(x^{12} - 16 x^{11} + 84 x^{10} + 376 x^{9} + 2042 x^{8} + 2400 x^{7} + 2832 x^{6} + 6240 x^{5} + 13628 x^{4} + 13184 x^{3} + 17552 x^{2} + 8672 x + 4056\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $33$ |
| Discriminant root field: | $\Q_{2}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[3, 4]$ |
Intermediate fields
| 2.3.0.1, 2.6.9.8 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{3} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 8 t^{2} x^{3} + \left(8 t^{2} + 4 t + 12\right) x^{2} + 8 x + 8 t^{2} + 20 t + 14 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[8, 4, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^4.(C_2\times A_4)$ (as 12T143) |
| Inertia group: | Intransitive group isomorphic to $C_2.D_4^2$ |
| Wild inertia group: | $C_2.D_4^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 3, 7/2, 7/2, 4]$ |
| Galois mean slope: | $231/64$ |
| Galois splitting model: |
$x^{12} - 84 x^{10} + 840 x^{8} + 27832 x^{6} - 659148 x^{4} + 5120304 x^{2} - 13832504$
|