Defining polynomial
\(x^{19} + 342 x^{18} + 2546\) |
Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $19$ |
Ramification exponent $e$: | $19$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $36$ |
Discriminant root field: | $\Q_{19}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 19 }) }$: | $19$ |
This field is Galois and abelian over $\Q_{19}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{19}$ |
Relative Eisenstein polynomial: | \( x^{19} + 342 x^{18} + 2546 \) |
Ramification polygon
Residual polynomials: | $z^{18} + 18$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[18, 0]$ |
Invariants of the Galois closure
Galois group: | $C_{19}$ (as 19T1) |
Inertia group: | $C_{19}$ (as 19T1) |
Wild inertia group: | $C_{19}$ |
Unramified degree: | $1$ |
Tame degree: | $1$ |
Wild slopes: | $[2]$ |
Galois mean slope: | $36/19$ |
Galois splitting model: | Not computed |