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
| $\Q_{7}(\sqrt{3})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7\cdot 3})$, 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: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | \( x^{6} - 7 t^{4} \)$\ \in\Q_{7}(t)[x]$ |