Defining polynomial
| \( x^{14} + 2 x^{12} + 2 x^{11} + 4 x^{10} + 2 x^{8} - 2 x^{7} - 2 x^{6} + 4 x^{4} + 4 x^{3} - 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $14$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $20$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-*})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-*})$, 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: | \( x^{14} + 2 x^{12} + 2 x^{11} + 4 x^{10} + 2 x^{8} - 2 x^{7} - 2 x^{6} + 4 x^{4} + 4 x^{3} - 2 \) |
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: | [2] |
| Galois mean slope: | $10/7$ |
| Galois splitting model: | $x^{14} - 49 x^{12} + 833 x^{10} - 5817 x^{8} - 44 x^{7} + 15771 x^{6} - 2156 x^{5} - 15323 x^{4} + 2156 x^{3} + 4963 x^{2} - 308 x - 23$ |