Defining polynomial
|
\(x^{14} + 4 x^{9} + 4 x^{8} + 4 x^{7} + 4 x^{4} + 4 x^{3} + 4 x + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $27$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[3]$ |
Intermediate fields
| 2.7.6.1 |
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: |
\( x^{14} + 4 x^{9} + 4 x^{8} + 4 x^{7} + 4 x^{4} + 4 x^{3} + 4 x + 2 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{12} + z^{10} + z^{8} + z^{6} + z^{4} + z^{2} + 1$ |
| Associated inertia: | $1$,$3$ |
| Indices of inseparability: | $[14, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\wr C_7:C_3$ (as 14T44) |
| Inertia group: | $C_2\wr C_7$ (as 14T29) |
| Wild inertia group: | $C_2^7$ |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | $[8/7, 8/7, 8/7, 20/7, 20/7, 20/7, 3]$ |
| Galois mean slope: | $1263/448$ |
| Galois splitting model: |
$x^{14} + 14 x^{12} + 14 x^{10} - 336 x^{8} - 784 x^{6} + 1890 x^{4} + 5670 x^{2} + 1458$
|