Defining polynomial
|
\(x^{14} + 4 x^{13} + 14 x^{12} + 656 x^{11} + 1236 x^{10} - 43472 x^{9} - 244456 x^{8} + 434816 x^{7} + 8570160 x^{6} + 35893184 x^{5} + 77018272 x^{4} + 105671936 x^{3} + 121134528 x^{2} + 74194176 x - 52979584\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $21$ |
| Discriminant root field: | $\Q_{2}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $14$ |
| This field is Galois and abelian over $\Q_{2}.$ | |
| Visible slopes: | $[3]$ |
Intermediate fields
| $\Q_{2}(\sqrt{2})$, 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: |
\( x^{2} + \left(4 t^{6} + 4 t^{3} + 4 t^{2} + 4\right) x + 8 t^{6} + 12 t^{5} + 4 t^{4} + 8 t^{3} + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_2$ |
| Unramified degree: | $7$ |
| Tame degree: | $1$ |
| Wild slopes: | $[3]$ |
| Galois mean slope: | $3/2$ |
| Galois splitting model: | $x^{14} + 84 x^{12} + 2156 x^{10} + 19208 x^{8} + 60368 x^{6} + 47040 x^{4} + 12544 x^{2} + 896$ |