Defining polynomial
| \( x^{14} - 56 x^{13} - 112 x^{12} + 49 x^{11} + 98 x^{10} - 147 x^{9} - 42 x^{7} + 98 x^{6} - 49 x^{5} + 98 x^{4} - 98 x^{3} + 147 x - 84 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $14$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $25$ |
| 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
| $\Q_{7}(\sqrt{7*})$, 7.7.12.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: | \( x^{14} - 56 x^{13} - 112 x^{12} + 49 x^{11} + 98 x^{10} - 147 x^{9} - 42 x^{7} + 98 x^{6} - 49 x^{5} + 98 x^{4} - 98 x^{3} + 147 x - 84 \) |