Defining polynomial
\(x^{14} + 1071 x^{13} + 491708 x^{12} + 125464437 x^{11} + 19221135969 x^{10} + 1769192651898 x^{9} + 90773424099277 x^{8} + 2022717194718445 x^{7} + 1543148215595651 x^{6} + 511298184911304 x^{5} + 94664842882197 x^{4} + 31823467716383 x^{3} + 1100435035211950 x^{2} + 24803053432496659 x + 29942903574449667\) |
Invariants
Base field: | $\Q_{17}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $7$ |
Discriminant exponent $c$: | $7$ |
Discriminant root field: | $\Q_{17}(\sqrt{17})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 17 }) }$: | $14$ |
This field is Galois and abelian over $\Q_{17}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{17}(\sqrt{17})$, 17.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 17.7.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{7} + 12 x + 14 \) |
Relative Eisenstein polynomial: | \( x^{2} + 153 x + 17 \) $\ \in\Q_{17}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{14}$ (as 14T1) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $7$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | Not computed |