Defining polynomial
| \( x^{14} + 7 x^{13} + 42 x^{12} + 7 x^{11} + 21 x^{9} + 14 x^{8} + 44 x^{7} + 35 x^{6} + 35 x^{5} + 21 x^{4} + 21 x^{3} + 21 x^{2} + 14 x + 3 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $7$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $16$ |
| Discriminant root field: | $\Q_{7}$ |
| 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(42 t + 7\right) x^{6} + \left(28 t + 35\right) x^{5} + \left(42 t + 7\right) x^{4} + \left(35 t + 14\right) x^{3} + \left(21 t + 21\right) x^{2} + 35 \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | 14T23 |
| Inertia group: | Intransitive group isomorphic to $C_7^2:C_3$ |
| Unramified degree: | $4$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3] |
| Galois mean slope: | $194/147$ |
| Galois splitting model: | $x^{14} - 14 x^{11} + 21 x^{10} + 21 x^{9} + 119 x^{8} - 248 x^{7} - 98 x^{6} - 392 x^{5} + 1001 x^{4} + 7 x^{3} + 7 x^{2} - 245 x - 211$ |