Defining polynomial
|
$( x^{2} + 6 x + 3 )^{7} + \left(28 x + 14\right) ( x^{2} + 6 x + 3 )^{5} + 7$
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{7})$: | $C_1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\frac{11}{6}]$ |
| Visible Swan slopes: | $[\frac{5}{6}]$ |
| Means: | $\langle\frac{5}{7}\rangle$ |
| Rams: | $(\frac{5}{6})$ |
| Jump set: | undefined |
| Roots of unity: | $48 = (7^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{2} + 6 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{7} + \left(21 t + 42\right) x^{5} + 7 \)
$\ \in\Q_{7}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + (3 t + 5)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
| Galois degree: | $588$ |
| Galois group: | $D_7:F_7$ (as 14T24) |
| Inertia group: | Intransitive group isomorphic to $C_7:F_7$ |
| Wild inertia group: | $C_7^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $6$ |
| Galois Artin slopes: | $[\frac{11}{6}, \frac{11}{6}]$ |
| Galois Swan slopes: | $[\frac{5}{6},\frac{5}{6}]$ |
| Galois mean slope: | $1.8129251700680271$ |
| Galois splitting model: |
$x^{14} - 42 x^{11} + 231 x^{10} - 602 x^{9} - 3234 x^{8} + 8831 x^{7} + 75509 x^{6} - 194922 x^{5} + 302603 x^{4} - 631687 x^{3} + 425859 x^{2} + 455441 x - 468537$
|