Defining polynomial
| \( x^{14} + 4 x^{12} + 4 x^{11} - 4 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{5} - 4 x^{4} + 6 x^{2} - 4 x - 6 \) |
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: | $i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-2*})$, 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: | \( x^{14} + 4 x^{12} + 4 x^{11} - 4 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{5} - 4 x^{4} + 6 x^{2} - 4 x - 6 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times C_7:C_3$ (as 14T5) |
| Inertia group: | $C_{14}$ |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | [3] |
| Galois mean slope: | $27/14$ |
| Galois splitting model: | $x^{14} + 42 x^{12} + 1288 x^{10} + 25760 x^{8} - 44 x^{7} + 378224 x^{6} + 9856 x^{5} + 3901408 x^{4} - 218064 x^{3} + 26367936 x^{2} + 692384 x + 90963044$ |