Defining polynomial
| \( x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $7$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $24$ |
| Discriminant root field: | $\Q_{7}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 7 })|$: | $14$ |
| This field is Galois and abelian over $\Q_{7}$. | |
Intermediate fields
| $\Q_{7}(\sqrt{*})$, 7.7.12.1 |
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(301 t + 203\right) x^{6} + \left(245 t + 147\right) x^{5} + \left(147 t + 245\right) x^{4} + \left(245 t + 49\right) x^{3} + \left(245 t + 98\right) x^{2} + \left(49 t + 196\right) x + 252 t + 280 \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | Intransitive group isomorphic to $C_7$ |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | [2] |
| Galois mean slope: | $12/7$ |
| Galois splitting model: | $x^{14} + 42 x^{12} + 623 x^{10} + 4431 x^{8} + 16513 x^{6} + 31906 x^{4} + 28784 x^{2} + 9409$ |