Defining polynomial
\(x^{8} + 38\) |
Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $7$ |
Discriminant root field: | $\Q_{19}(\sqrt{19\cdot 2})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 19 }) }$: | $2$ |
This field is not Galois over $\Q_{19}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{19}(\sqrt{19})$, 19.4.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{19}$ |
Relative Eisenstein polynomial: | \( x^{8} + 38 \) |
Ramification polygon
Residual polynomials: | $z^{7} + 8z^{6} + 9z^{5} + 18z^{4} + 13z^{3} + 18z^{2} + 9z + 8$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $\SD_{16}$ (as 8T8) |
Inertia group: | $C_8$ (as 8T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $8$ |
Wild slopes: | None |
Galois mean slope: | $7/8$ |
Galois splitting model: | Not computed |