Defining polynomial
| \( x^{14} + 2 x^{13} + 2 x^{12} + 4 x^{11} + 4 x^{9} + 4 x^{8} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 4 x - 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $14$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $26$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
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: | \( x^{14} + 2 x^{13} + 2 x^{12} + 4 x^{11} + 4 x^{9} + 4 x^{8} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 4 x - 2 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times F_8:C_3$ (as 14T18) |
| Inertia group: | 14T9 |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | [2, 20/7, 20/7, 20/7] |
| Galois mean slope: | $75/28$ |
| Galois splitting model: | $x^{14} - 7 x^{12} - 245 x^{10} + 2107 x^{8} + 7203 x^{6} - 69237 x^{4} + 27783 x^{2} + 3969$ |