Defining polynomial
| \( x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $4$ |
| Discriminant exponent $c$ : | $8$ |
| Discriminant root field: | $\Q_{7}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 7 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{7}$. | |
Intermediate fields
| $\Q_{7}(\sqrt{*})$, 7.3.2.2, 7.4.0.1, 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: | 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^{3} \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: | $x^{12} - x^{11} + 3 x^{10} - 4 x^{9} + 9 x^{8} + 2 x^{7} + 12 x^{6} + x^{5} + 25 x^{4} - 11 x^{3} + 5 x^{2} - 2 x + 1$ |