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 |