Defining polynomial
\(x^{16} + 16 x^{15} + 16 x^{13} + 16 x^{9} + 20 x^{8} + 16 x^{6} + 24 x^{4} + 2\) |
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: | $[3, 4, 5, 41/8]$ |
Intermediate fields
$\Q_{2}(\sqrt{2})$, 2.4.11.16, 2.8.31.188 |
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: | \( x^{16} + 16 x^{15} + 16 x^{13} + 16 x^{9} + 20 x^{8} + 16 x^{6} + 24 x^{4} + 2 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{4} + 1$,$z^{8} + 1$ |
Associated inertia: | $1$,$1$,$1$,$1$ |
Indices of inseparability: | $[57, 48, 32, 16, 0]$ |