Defining polynomial
\(x^{16} + 24 x^{8} + 32 x^{5} + 40 x^{4} + 32 x^{2} + 34\) |
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: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[3, 4, 5, 6]$ |
Intermediate fields
$\Q_{2}(\sqrt{-2})$, 2.4.11.17, 2.8.31.69 |
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: | \( x^{16} + 24 x^{8} + 32 x^{5} + 40 x^{4} + 32 x^{2} + 34 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{4} + 1$,$z^{8} + 1$ |
Associated inertia: | $1$,$1$,$1$,$1$ |
Indices of inseparability: | $[64, 48, 32, 16, 0]$ |
Invariants of the Galois closure
Galois group: | $D_4^2.D_4$ (as 16T1005) |
Inertia group: | $D_4^2:C_4$ (as 16T663) |
Wild inertia group: | $D_4^2:C_4$ |
Unramified degree: | $2$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 3, 7/2, 4, 17/4, 5, 21/4, 6]$ |
Galois mean slope: | $693/128$ |
Galois splitting model: | $x^{16} + 72 x^{12} + 144 x^{10} + 1152 x^{8} + 6912 x^{6} + 11448 x^{4} + 28512 x^{2} + 58482$ |