Defining polynomial
|
\(x^{14} - 756 x^{13} + 407484 x^{12} - 120759324 x^{11} + 18245133144 x^{10} - 1371646043208 x^{9} + 45234910589400 x^{8} - 365498518846944 x^{7} - 160150981540032 x^{6} - 9636001680480 x^{5} - 1806831694584 x^{4} - 187090145760 x^{3} - 8419197738 x^{2} - 138355224 x - 3294172\)
|
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: | $i$ |
| $\card{ \Gal(K/\Q_{ 7 }) }$: | $14$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
| Visible slopes: | None |
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
\( x^{7} + 6 x + 4 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + \left(21 t^{6} + 42 t^{4} + 42 t^{3} + 42 t^{2} + 42 t\right) x + 7 t \)
$\ \in\Q_{7}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |