Defining polynomial
|
\(x^{14} + 42 x^{12} + 7\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $25$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ |
| Root number: | $-i$ |
| $\card{ \Gal(K/\Q_{ 7 }) }$: | $14$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| $\Q_{7}(\sqrt{7\cdot 3})$, 7.7.12.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 42 x^{12} + 7 \)
|
Ramification polygon
| Residual polynomials: | $2z^{6} + 5$,$z^{7} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[12, 0]$ |