Defining polynomial
|
\(x^{8} + 4 x^{7} + 4 x^{4} + 8 x^{2} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[3, 19/6, 19/6]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-2})$ |
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: |
\( x^{8} + 4 x^{7} + 4 x^{4} + 8 x^{2} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{4} + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[15, 15, 8, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^3:S_4$ (as 8T41) |
| Inertia group: | $C_2^3:A_4$ (as 8T33) |
| Wild inertia group: | $C_2^2\wr C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[4/3, 4/3, 3, 19/6, 19/6]$ |
| Galois mean slope: | $139/48$ |
| Galois splitting model: | $x^{8} - 4 x^{7} + 16 x^{6} - 16 x^{5} + 32 x^{4} + 32 x^{3} + 16 x^{2} + 56 x + 54$ |