Defining polynomial
|
\(x^{12} + 10 x^{11} + 51 x^{10} + 176 x^{9} + 450 x^{8} + 870 x^{7} + 1299 x^{6} + 1516 x^{5} + 1250 x^{4} + 542 x^{3} + 67 x^{2} - 56 x + 7\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $20$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $4$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[8/3]$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.3.2.1 x3, 2.6.4.1, 2.6.10.2, 2.6.10.3 |
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: |
\( x^{6} + 2 x^{5} + 2 x^{4} + 4 x^{3} + 2 x^{2} + 6 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{4} + z^{2} + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
| Galois group: | $S_4$ (as 12T9) |
| Inertia group: | Intransitive group isomorphic to $A_4$ |
| Wild inertia group: | $C_2^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[8/3, 8/3]$ |
| Galois mean slope: | $13/6$ |
| Galois splitting model: | $x^{12} + 3 x^{10} - 5 x^{6} + 3 x^{2} + 1$ |