Defining polynomial
|
\(x^{14} + 14 x^{13} + 126 x^{12} + 784 x^{11} + 4300 x^{10} + 19592 x^{9} + 80680 x^{8} + 276608 x^{7} + 822832 x^{6} + 1982880 x^{5} + 3998112 x^{4} + 6222080 x^{3} + 7679040 x^{2} + 6275456 x + 3453824\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $-i$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $14$ |
| This field is Galois and abelian over $\Q_{2}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-5})$, 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: |
\( x^{2} + 2 x + 4 t^{5} + 4 t^{4} + 4 t^{3} + 4 t^{2} + 4 t + 6 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |