Defining polynomial
\(x^{12} + 56 x^{6} + 1323\)  |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 7 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
Intermediate fields
| $\Q_{7}(\sqrt{3})$, 7.3.2.2, 7.4.2.2, 7.6.4.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | \( x^{6} - 7 t^{3} \)$\ \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{12}$ (as 12T1) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | $x^{12} - x^{11} - 12 x^{10} + 11 x^{9} + 54 x^{8} - 43 x^{7} - 113 x^{6} + 71 x^{5} + 110 x^{4} - 46 x^{3} - 40 x^{2} + 8 x + 1$ |