Defining polynomial
\(x^{6} + 2 x^{3} + 6\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
$\Q_{2}(\sqrt{-5})$, 2.3.2.1 |
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: | \( x^{6} + 2 x^{3} + 6 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{4} + z^{2} + 1$ |
Associated inertia: | $1$,$2$ |
Indices of inseparability: | $[3, 0]$ |