Defining polynomial
|
\(x^{12} + 996 x^{11} + 413370 x^{10} + 91510820 x^{9} + 11398264215 x^{8} + 757668341256 x^{7} + 21038145869850 x^{6} + 3788341872612 x^{5} + 285025608105 x^{4} + 26708684540 x^{3} + 1901007431580 x^{2} + 126186836966352 x + 3490535486974976\)
|
Invariants
| Base field: | $\Q_{167}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{167}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 167 }) }$: | $12$ |
| This field is Galois over $\Q_{167}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{167}(\sqrt{5})$, $\Q_{167}(\sqrt{167})$, $\Q_{167}(\sqrt{167\cdot 5})$, 167.3.2.1 x3, 167.4.2.1, 167.6.4.1, 167.6.5.1 x3, 167.6.5.2 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{167}(\sqrt{5})$ $\cong \Q_{167}(t)$ where $t$ is a root of
\( x^{2} + 166 x + 5 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + 167 \)
$\ \in\Q_{167}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{5} + 6z^{4} + 15z^{3} + 20z^{2} + 15z + 6$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_6$ (as 12T3) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | Not computed |