Defining polynomial
|
\(x^{12} + 21\)
|
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: | $i$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $6$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | None |
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: |
\( x^{12} + 21 \)
|
Ramification polygon
| Residual polynomials: | $z^{11} + 5z^{10} + 3z^{9} + 3z^{8} + 5z^{7} + z^{6} + z^{4} + 5z^{3} + 3z^{2} + 3z + 5$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | $C_{12}$ (as 12T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | None |
| Galois mean slope: | $11/12$ |
| Galois splitting model: |
$x^{12} - 42 x^{10} - 56 x^{9} + 315 x^{8} + 336 x^{7} - 2870 x^{6} - 7056 x^{5} - 2079 x^{4} + 7896 x^{3} + 7056 x^{2} - 1792$
|