Defining polynomial
\(x^{14} + 2 x^{13} + 70 x^{12} + 808 x^{11} + 7772 x^{10} + 51528 x^{9} + 259720 x^{8} + 1006080 x^{7} + 3047152 x^{6} + 7154528 x^{5} + 12862880 x^{4} + 17283456 x^{3} + 16442432 x^{2} + 9668992 x + 2866048\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $7$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
2.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 2.7.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{7} + x + 1 \) |
Relative Eisenstein polynomial: | \( x^{2} + \left(2 t^{6} + 2 t^{5} + 2 t^{4} + 2 t^{2} + 2\right) x + 4 t^{6} + 4 t^{5} + 4 t^{4} + 4 t^{3} + 4 t^{2} + 6 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + t^{6} + t^{5} + t^{4} + t^{2} + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\wr C_7$ (as 14T29) |
Inertia group: | Intransitive group isomorphic to $C_2^7$ |
Wild inertia group: | $C_2^7$ |
Unramified degree: | $7$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2, 2, 2, 2, 2, 2]$ |
Galois mean slope: | $127/64$ |
Galois splitting model: | $x^{14} - 14 x^{12} + 42 x^{10} + 119 x^{8} - 490 x^{6} - 364 x^{4} + 1246 x^{2} + 961$ |