Defining polynomial
|
\(x^{12} - 4 x^{11} + 20 x^{10} - 64 x^{9} + 196 x^{8} - 336 x^{7} + 736 x^{6} + 160 x^{5} + 1712 x^{4} + 5632 x^{3} + 5440 x^{2} + 6784 x + 5824\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.3.0.1, 2.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{6} + x^{4} + x^{3} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 2 t^{2} x + 4 t^{5} + 4 t^{3} + 4 t^{2} + 6 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + t^{2}$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_4\times A_4$ (as 12T51) |
| Inertia group: | Intransitive group isomorphic to $C_2^4$ |
| Wild inertia group: | $C_2^4$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 2]$ |
| Galois mean slope: | $15/8$ |
| Galois splitting model: | $x^{12} - 6 x^{10} + 15 x^{8} - 21 x^{6} + 18 x^{4} - 9 x^{2} + 3$ |