Defining polynomial
|
\(x^{16} - 198 x^{8} - 10043\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $16$ |
| Ramification exponent $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{11}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 11 }) }$: | $16$ |
| This field is Galois over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{11}(\sqrt{2})$, $\Q_{11}(\sqrt{11})$, $\Q_{11}(\sqrt{11\cdot 2})$, 11.4.2.1, 11.4.3.1 x2, 11.4.3.2 x2, 11.8.6.2, 11.8.7.2 x4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{2} + 7 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{8} + 44 t + 55 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{7} + 8z^{6} + 6z^{5} + z^{4} + 4z^{3} + z^{2} + 6z + 8$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $\SD_{16}$ (as 16T12) |
| Inertia group: | Intransitive group isomorphic to $C_8$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $8$ |
| Wild slopes: | None |
| Galois mean slope: | $7/8$ |
| Galois splitting model: | Not computed |