Defining polynomial
|
\(x^{12} + 12 x^{11} + 40 x^{10} - 8 x^{9} - 58 x^{8} + 552 x^{7} + 640 x^{6} + 1328 x^{5} - 12 x^{4} + 1760 x^{3} - 720 x^{2} + 576 x - 104\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2, 3]$ |
Intermediate fields
| 2.3.0.1, 2.6.6.6 |
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: |
\( x^{4} + 4 x^{3} + \left(4 t^{2} + 6 t\right) x^{2} + 4 t^{2} x + 4 t^{2} + 12 t + 6 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $tz + t^{2}$,$z^{2} + t$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_4^3:C_6$ (as 12T141) |
| Inertia group: | Intransitive group isomorphic to $C_4^2:C_2^2$ |
| Wild inertia group: | $C_4^2:C_2^2$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 2, 3, 3]$ |
| Galois mean slope: | $87/32$ |
| Galois splitting model: | $x^{12} - 6 x^{10} + 12 x^{8} + 6 x^{6} - 27 x^{4} - 36 x^{2} - 3$ |