Defining polynomial
\(x^{14} + 2407 x^{7} + 1839267\)  |
Invariants
| Base field: | $\Q_{29}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{29}(\sqrt{2})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 29 })|$: | $14$ |
| This field is Galois and abelian over $\Q_{29}.$ | |
Intermediate fields
| $\Q_{29}(\sqrt{2})$, 29.7.6.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{29}(\sqrt{2})$ $\cong \Q_{29}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | \( x^{7} - 29 t^{7} \)$\ \in\Q_{29}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | Intransitive group isomorphic to $C_7$ |
| Unramified degree: | $2$ |
| Tame degree: | $7$ |
| Wild slopes: | None |
| Galois mean slope: | $6/7$ |
| Galois splitting model: | Not computed |