Defining polynomial
| \( x^{14} + 7 x^{13} + 35 x^{12} + 7 x^{11} + 42 x^{10} + 42 x^{9} + 42 x^{8} + 43 x^{7} + 14 x^{6} + 35 x^{5} + 14 x^{4} + 7 x + 3 \) |
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: | $1$ |
| $|\Aut(K/\Q_{ 7 })|$: | $1$ |
| This field is not Galois over $\Q_{7}$. | |
Intermediate fields
| $\Q_{7}(\sqrt{*})$ |
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} + 14 t x^{5} + 7 x^{4} + \left(21 t + 7\right) x^{3} + \left(14 t + 35\right) x^{2} + 35 x + 28 t + 28 \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | 14T24 |
| Inertia group: | Intransitive group isomorphic to $C_7^2:C_3:C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | [7/6, 7/6] |
| Galois mean slope: | $341/294$ |
| Galois splitting model: | $x^{14} - 42 x^{11} + 21 x^{10} + 42 x^{9} + 63 x^{8} + 390 x^{7} + 147 x^{6} + 294 x^{5} - 1764 x^{4} - 3045 x^{3} - 1043 x^{2} - 2919 x - 3141$ |