Defining polynomial
|
\(x^{12} + 4 x^{10} + 4 x^{9} + 4 x^{8} + 12 x^{6} + 8 x^{5} + 8 x^{4} + 10\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $32$ |
| 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: | $[3, 7/2]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-2\cdot 5})$, 2.3.2.1, 2.4.10.6, 2.6.11.13 |
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: |
\( x^{12} + 4 x^{10} + 4 x^{9} + 4 x^{8} + 12 x^{6} + 8 x^{5} + 8 x^{4} + 10 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{8} + z^{4} + 1$ |
| Associated inertia: | $1$,$1$,$2$ |
| Indices of inseparability: | $[21, 12, 0]$ |