Defining polynomial
\(x^{14} - 9082 x^{13} + 74145923 x^{12} + 132432948192 x^{11} + 1786839611560128 x^{10} + 5067376315465156928 x^{9} + 9248554083250091316158 x^{8} + 5246110013130072565373868 x^{7} + 175722527581751735007002 x^{6} + 1829322849882921651008 x^{5} + 12255932895690917952 x^{4} + 17258794241329632 x^{3} + 183592645794377 x^{2} - 427270691242 x + 893871739\) |
Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $7$ |
Discriminant exponent $c$: | $7$ |
Discriminant root field: | $\Q_{19}(\sqrt{19\cdot 2})$ |
Root number: | $-i$ |
$\card{ \Gal(K/\Q_{ 19 }) }$: | $14$ |
This field is Galois and abelian over $\Q_{19}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{19}(\sqrt{19\cdot 2})$, 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: | \( x^{2} + \left(304 t^{6} + 342 t^{5} + 285 t^{4} + 304 t^{3} + 247 t^{2} + 323 t + 266\right) x + 19 \) $\ \in\Q_{19}(t)[x]$ |
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 |