Defining polynomial
| \( x^{14} + 7 x^{13} + 7 x^{11} + 7 x^{10} + 42 x^{9} + 14 x^{8} + 22 x^{7} + 35 x^{6} + 21 x^{4} + 7 x^{3} + 7 x^{2} + 42 x + 17 \) |
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
| $\Q_{7}(\sqrt{*})$, 7.7.7.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | $ x^{7} + \left(28 t + 42\right) x^{6} + \left(28 t + 7\right) x^{5} + \left(42 t + 42\right) x^{4} + \left(21 t + 21\right) x^{3} + 14 t x^{2} + \left(35 t + 42\right) x + 35 t + 35 \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: | $x^{14} + 84 x^{12} + 2772 x^{10} + 45360 x^{8} + 381024 x^{6} + 1524096 x^{4} + 2286144 x^{2} + 1399680$ |