Defining polynomial
|
\(x^{12} - 4 x^{11} + 16 x^{10} - 4 x^{9} + 32 x^{8} + 32 x^{7} + 96 x^{5} - 4 x^{4} + 72 x^{3} + 88 x^{2} + 108\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $12$ |
| This field is Galois over $\Q_{2}.$ | |
| Visible slopes: | $[3]$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.3.2.1 x3, 2.4.6.2, 2.6.4.1, 2.6.11.9 x3, 2.6.11.13 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + 4 t x^{5} + 4 t x^{3} + \left(4 t + 2\right) x^{2} + 4 x + 12 t + 6 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{4} + z^{2} + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_6$ (as 12T3) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[3]$ |
| Galois mean slope: | $11/6$ |
| Galois splitting model: | $x^{12} + 2 x^{10} + 4 x^{8} - 4 x^{6} + 20 x^{4} - 16 x^{2} + 4$ |