Defining polynomial
| \( x^{14} - 686 x^{8} + 117649 x^{2} - 3294172 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $7$ |
| Discriminant exponent $c$ : | $7$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7})$ |
| Root number: | $i$ |
| $|\Gal(K/\Q_{ 7 })|$: | $14$ |
| This field is Galois and abelian over $\Q_{7}$. | |
Intermediate fields
| $\Q_{7}(\sqrt{7})$, 7.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 7.7.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{7} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{2} + \left(42 t^{6} + 35 t^{5} + 28 t^{4} + 42 t^{3} + 21 t^{2} + 28\right) x + 35 t^{4} + 35 t^{3} + 14 t^{2} + 21 t + 42 \in\Q_{7}(t)[x]$ |