Defining polynomial
|
\(x^{12} - 112096 x^{6} - 1542341972\)
|
Invariants
| Base field: | $\Q_{113}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{113}$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 113 }) }$: | $6$ |
| This field is not Galois over $\Q_{113}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{113}(\sqrt{3})$, $\Q_{113}(\sqrt{113})$, $\Q_{113}(\sqrt{113\cdot 3})$, 113.4.2.1, 113.6.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{113}(\sqrt{3})$ $\cong \Q_{113}(t)$ where $t$ is a root of
\( x^{2} + 101 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + 1356 t + 12430 \)
$\ \in\Q_{113}(t)[x]$
|
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: | $C_6\times S_3$ (as 12T18) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | Not computed |