Defining polynomial
| \( x^{14} + 28 x^{13} + 21 x^{12} + 42 x^{11} + 14 x^{10} + 28 x^{8} + 37 x^{7} + 28 x^{5} + 7 x^{4} + 7 x^{3} + 35 x^{2} + 28 x + 31 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $7$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $16$ |
| Discriminant root field: | $\Q_{7}(\sqrt{*})$ |
| 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} + 7 t x^{6} + \left(21 t + 42\right) x^{5} + \left(21 t + 28\right) x^{4} + \left(21 t + 35\right) x^{3} + \left(7 t + 21\right) x^{2} + 42 t + 21 \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_7^2:C_6$ (as 14T14) |
| Inertia group: | Intransitive group isomorphic to $C_7^2:C_3$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3] |
| Galois mean slope: | $194/147$ |
| Galois splitting model: | $x^{14} - 84 x^{12} - 84 x^{11} + 3206 x^{10} + 4956 x^{9} - 66668 x^{8} - 130154 x^{7} + 786842 x^{6} + 1801856 x^{5} - 4775288 x^{4} - 13285916 x^{3} + 9985696 x^{2} + 39242252 x + 21880010$ |