Defining polynomial
\(x^{14} - 12 x^{13} + 142 x^{12} - 640 x^{11} + 2548 x^{10} - 3904 x^{9} + 12280 x^{8} - 34048 x^{7} + 159536 x^{6} - 121792 x^{5} - 109152 x^{4} - 121344 x^{3} + 389056 x^{2} - 197120 x + 9856\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $7$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
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: | \( x^{2} + \left(2 t^{6} + 2 t^{4} + 2 t^{3} + 2 t\right) x + 4 t^{5} + 4 t^{2} + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + t^{6} + t^{4} + t^{3} + t$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $F_8$ (as 14T6) |
Inertia group: | Intransitive group isomorphic to $C_2^3$ |
Wild inertia group: | $C_2^3$ |
Unramified degree: | $7$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2, 2]$ |
Galois mean slope: | $7/4$ |
Galois splitting model: | $x^{14} - 84 x^{10} + 70 x^{8} + 1505 x^{6} - 7 x^{4} - 6405 x^{2} - 4489$ |