Defining polynomial
\(x^{12} - 14 x^{8} + 49 x^{4} - 1372\)  |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 7 })|$: | $6$ |
| This field is not Galois over $\Q_{7}.$ | |
Intermediate fields
| $\Q_{7}(\sqrt{7})$, 7.3.0.1, 7.4.3.2, 7.6.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 7.3.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{3} - x + 2 \) |
| Relative Eisenstein polynomial: | \( x^{4} - 7 t^{2} \)$\ \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Unramified degree: | $6$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | $x^{12} - 6 x^{11} + 15 x^{10} - 16 x^{9} + 12 x^{8} - 66 x^{7} - 67 x^{6} + 942 x^{5} - 3261 x^{4} + 4534 x^{3} - 408 x^{2} - 1692 x - 159$ |