Defining polynomial
|
\(x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 12 x^{12} + 16 x^{9} + 12 x^{8} + 8 x^{4} + 48 x^{2} + 50\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification exponent $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $72$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[3, 7/2, 19/4, 43/8]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-2})$, 2.4.10.8, 2.8.29.128 |
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: |
\( x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 12 x^{12} + 16 x^{9} + 12 x^{8} + 8 x^{4} + 48 x^{2} + 50 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{4} + 1$,$z^{8} + 1$ |
| Associated inertia: | $1$,$1$,$1$,$1$ |
| Indices of inseparability: | $[57, 44, 28, 16, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^4.C_2\wr D_4$ (as 16T1473) |
| Inertia group: | $C_2^4.C_2\wr C_4$ (as 16T1135) |
| Wild inertia group: | data not computed |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8]$ |
| Galois mean slope: | $2627/512$ |
| Galois splitting model: | $x^{16} - 8 x^{14} + 48 x^{12} - 240 x^{10} + 612 x^{8} - 720 x^{6} + 864 x^{4} - 864 x^{2} + 324$ |