Defining polynomial
|
\(x^{16} + 16 x^{14} + 40 x^{12} + 32 x^{11} + 16 x^{10} + 4 x^{8} + 32 x^{5} + 32 x^{4} + 32 x^{2} + 34\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification exponent $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $79$ |
| Discriminant root field: | $\Q_{2}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $8$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[3, 4, 5, 6]$ |
Intermediate fields
| $\Q_{2}(\sqrt{2})$, 2.4.11.1, 2.8.31.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{16} + 16 x^{14} + 40 x^{12} + 32 x^{11} + 16 x^{10} + 4 x^{8} + 32 x^{5} + 32 x^{4} + 32 x^{2} + 34 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{4} + 1$,$z^{8} + 1$ |
| Associated inertia: | $1$,$1$,$1$,$1$ |
| Indices of inseparability: | $[64, 48, 32, 16, 0]$ |