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: | $\Q_{11}(\sqrt{*})$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 11 })|$: | $12$ |
| This field is Galois over $\Q_{11}$. | |
Intermediate fields
| $\Q_{11}(\sqrt{*})$, 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: | $\Q_{11}(\sqrt{*})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} - x + 7 \) |
| Relative Eisenstein polynomial: | $ x^{6} - 11 t^{3} \in\Q_{11}(t)[x]$ |