Defining polynomial
| \( x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 7 })|$: | $8$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
Intermediate fields
| $\Q_{7}(\sqrt{3})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7\cdot 3})$, 7.4.0.1, 7.4.2.1, 7.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 7.4.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{4} + x^{2} - 3 x + 5 \) |
| Relative Eisenstein polynomial: | $ x^{2} - 7 t^{2} \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_4$ (as 8T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 6 x^{3} - 4 x^{2} - 8 x + 16$ |