Defining polynomial
|
\(x^{14} + 28 x^{2} + 7\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{7})$: | $C_2$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\frac{7}{6}]$ |
| Visible Swan slopes: | $[\frac{1}{6}]$ |
| Means: | $\langle\frac{1}{7}\rangle$ |
| Rams: | $(\frac{1}{3})$ |
| Jump set: | undefined |
| Roots of unity: | $6 = (7 - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{7\cdot 3})$, 7.1.7.7a1.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 28 x^{2} + 7 \)
|
Ramification polygon
| Residual polynomials: | $z^7 + 2$,$2 z^2 + 6$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $42$ |
| Galois group: | $F_7$ (as 14T4) |
| Inertia group: | $F_7$ (as 14T4) |
| Wild inertia group: | $C_7$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $6$ |
| Galois Artin slopes: | $[\frac{7}{6}]$ |
| Galois Swan slopes: | $[\frac{1}{6}]$ |
| Galois mean slope: | $1.119047619047619$ |
| Galois splitting model: |
$x^{14} - 28 x^{12} + 112 x^{10} + 1645 x^{8} + 5530 x^{6} + 9163 x^{4} + 7721 x^{2} + 2800$
|