Defining polynomial
| \( x^{12} - 49 x^{6} + 3969 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $10$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 7 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{7}$. | |
Intermediate fields
| $\Q_{7}(\sqrt{*})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7*})$, 7.3.2.1, 7.4.2.1, 7.6.4.2, 7.6.5.3, 7.6.5.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | $ x^{6} - 7 t^{4} \in\Q_{7}(t)[x]$ |