Defining polynomial
|
\(x^{14} - 836 x^{13} + 11529618 x^{12} + 40153059632 x^{11} + 45701018933715 x^{10} - 27199543314328911 x^{9} - 122971941784309357645 x^{8} - 171679318980585845579840 x^{7} - 4585575958956710706912 x^{6} + 50052189831457642637 x^{5} + 1651410192170735556 x^{4} - 8261685479759434 x^{3} - 232742465638578 x^{2} - 829512973792 x - 15195819563\)
|
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $7$ |
| Discriminant root field: | $\Q_{19}(\sqrt{19})$ |
| Root number: | $i$ |
| $\card{ \Gal(K/\Q_{ 19 }) }$: | $14$ |
| This field is Galois and abelian over $\Q_{19}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{19}(\sqrt{19})$, 19.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 19.7.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of
\( x^{7} + 6 x + 17 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + \left(38 t^{6} + 266 t^{5} + 19 t^{4} + 304 t^{3} + 209 t^{2} + 152 t + 76\right) x + 19 t \)
$\ \in\Q_{19}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $7$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |