Defining polynomial
| \( x^{12} + 49 x^{6} - 1029 x^{3} + 12005 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| 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
| $\Q_{7}(\sqrt{3})$, 7.3.2.1, 7.4.0.1, 7.6.4.2 |
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^{3} - 7 t \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{12}$ (as 12T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Unramified degree: | $4$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | $x^{12} - x^{11} + x^{10} - 27 x^{9} + 27 x^{8} + 90 x^{7} + 53 x^{6} - 1353 x^{5} + 768 x^{4} + 3886 x^{3} + 1600 x^{2} - 5409 x + 1847$ |