Defining polynomial
\(x^{14} - 56 x^{9} + 56 x^{8} + 14 x^{7} - 392 x^{4} - 1568 x^{3} + 392 x^{2} + 392 x + 49\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 7 }) }$: | $2$ |
This field is not Galois over $\Q_{7}.$ | |
Visible slopes: | $[7/6]$ |
Intermediate fields
$\Q_{7}(\sqrt{3})$, 7.7.7.6 |
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 \( x^{2} + 6 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{7} + \left(14 t + 14\right) x^{2} + 28 x + 7 \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\times F_7$ (as 14T7) |
Inertia group: | Intransitive group isomorphic to $F_7$ |
Wild inertia group: | $C_7$ |
Unramified degree: | $2$ |
Tame degree: | $6$ |
Wild slopes: | $[7/6]$ |
Galois mean slope: | $47/42$ |
Galois splitting model: | $x^{14} + 42 x^{12} + 693 x^{10} + 9856 x^{8} - 26460 x^{6} + 41328 x^{4} - 44352 x^{2} + 20736$ |