Defining polynomial
|
\(x^{12} - 165 x^{6} - 4356\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 11 }) }$: | $12$ |
| This field is Galois over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{11}(\sqrt{2})$, 11.3.2.1 x3, 11.4.2.2, 11.6.4.1 |
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: |
\( x^{6} + 33 t + 33 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{5} + 6z^{4} + 4z^{3} + 9z^{2} + 4z + 6$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |