Defining polynomial
| \( x^{14} + 35 x^{13} + 21 x^{12} + 28 x^{11} + 7 x^{10} + 7 x^{9} + 43 x^{7} + 28 x^{6} + 28 x^{5} + 21 x^{4} + 7 x^{2} + 42 x + 10 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $7$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $14$ |
| Discriminant root field: | $\Q_{7}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 7 })|$: | $2$ |
| This field is not Galois over $\Q_{7}$. | |
Intermediate fields
| $\Q_{7}(\sqrt{*})$, 7.7.7.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | $ x^{7} + \left(14 t + 35\right) x^{6} + \left(14 t + 14\right) x^{5} + \left(21 t + 14\right) x^{4} + \left(14 t + 21\right) x^{3} + \left(21 t + 21\right) x^{2} + \left(28 t + 35\right) x + 7 t + 14 \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times F_7$ (as 14T7) |
| Inertia group: | Intransitive group isomorphic to $F_7$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | [7/6] |
| Galois mean slope: | $47/42$ |
| Galois splitting model: | $x^{14} - 14 x^{11} - 42 x^{10} - 84 x^{9} - 280 x^{8} - 1338 x^{7} - 4116 x^{6} - 11858 x^{5} - 25158 x^{4} - 39900 x^{3} - 50057 x^{2} - 46326 x - 17244$ |