Defining polynomial
| \( x^{12} - 28 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $11$ |
| 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
| $\Q_{7}(\sqrt{7})$, 7.3.2.3, 7.4.3.2, 7.6.5.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: | \( x^{12} - 28 \) |
Invariants of the Galois closure
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | $C_{12}$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | None |
| Galois mean slope: | $11/12$ |
| Galois splitting model: | $x^{12} - 42 x^{10} - 56 x^{9} + 315 x^{8} + 336 x^{7} - 2870 x^{6} - 7056 x^{5} - 2079 x^{4} + 7896 x^{3} + 7056 x^{2} - 1792$ |