Defining polynomial
|
\(x^{14} + 126 x^{13} + 6818 x^{12} + 205632 x^{11} + 3742284 x^{10} + 41321448 x^{9} + 260402968 x^{8} + 775862822 x^{7} + 520808330 x^{6} + 165413472 x^{5} + 33744732 x^{4} + 71380792 x^{3} + 731029768 x^{2} + 4357641344 x + 11135742105\)
|
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{19}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 19 }) }$: | $2$ |
| This field is not Galois over $\Q_{19}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{19}(\sqrt{2})$, 19.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{19}(\sqrt{2})$ $\cong \Q_{19}(t)$ where $t$ is a root of
\( x^{2} + 18 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{7} + 19 \)
$\ \in\Q_{19}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{6} + 7z^{5} + 2z^{4} + 16z^{3} + 16z^{2} + 2z + 7$ |
| Associated inertia: | $3$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $F_7$ (as 14T4) |
| Inertia group: | Intransitive group isomorphic to $C_7$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $7$ |
| Wild slopes: | None |
| Galois mean slope: | $6/7$ |
| Galois splitting model: | Not computed |