Defining polynomial
\(x^{14} - 1716 x^{7} - 91260\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 13 }) }$: | $7$ |
This field is not Galois over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{13}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{13}(\sqrt{2})$ $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{2} + 12 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{7} + 156 t + 78 \) $\ \in\Q_{13}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{6} + 7z^{5} + 8z^{4} + 9z^{3} + 9z^{2} + 8z + 7$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_7\times D_7$ (as 14T8) |
Inertia group: | Intransitive group isomorphic to $C_7$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $14$ |
Tame degree: | $7$ |
Wild slopes: | None |
Galois mean slope: | $6/7$ |
Galois splitting model: | Not computed |