Defining polynomial
\(x^{12} + 220 x^{6} + 41503\)  |
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: | $-1$ |
| $|\Gal(K/\Q_{ 11 })|$: | $12$ |
| This field is Galois over $\Q_{11}.$ | |
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 \( x^{2} - x + 7 \) |
| Relative Eisenstein polynomial: | \( x^{6} - 11 t^{3} \)$\ \in\Q_{11}(t)[x]$ |