Defining polynomial
|
\(x^{14} + 158 x^{12} + 1568 x^{11} + 1332 x^{10} - 32384 x^{9} + 95128 x^{8} + 683008 x^{7} + 3694128 x^{6} + 18752000 x^{5} + 49924512 x^{4} + 51764736 x^{3} - 10775616 x^{2} + 27525120 x + 167639168\)
|
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: | $-i$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $14$ |
| 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: |
\( x^{2} + \left(4 t^{5} + 4 t^{3} + 4 t^{2}\right) x + 12 t^{6} + 8 t^{4} + 4 t^{3} + 8 t^{2} + 8 t + 10 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |