Defining polynomial
| \( x^{14} + 2 x^{12} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{4} + 2 x^{3} + 2 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $14$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $14$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| 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: | \( x^{14} + 2 x^{12} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{4} + 2 x^{3} + 2 x + 2 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times F_8:C_3$ (as 14T18) |
| Inertia group: | 14T6 |
| Unramified degree: | $6$ |
| Tame degree: | $7$ |
| Wild slopes: | [8/7, 8/7, 8/7] |
| Galois mean slope: | $31/28$ |
| Galois splitting model: | $x^{14} - 35 x^{12} - 42 x^{11} + 371 x^{10} + 1218 x^{9} - 1617 x^{8} - 10412 x^{7} + 2247 x^{6} + 38108 x^{5} - 469 x^{4} - 69034 x^{3} + 10885 x^{2} + 38402 x - 3623$ |