Defining polynomial
\(x^{22} + 394\) |
Invariants
Base field: | $\Q_{197}$ |
Degree $d$: | $22$ |
Ramification exponent $e$: | $22$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $21$ |
Discriminant root field: | $\Q_{197}(\sqrt{197\cdot 2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 197 }) }$: | $2$ |
This field is not Galois over $\Q_{197}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{197}(\sqrt{197\cdot 2})$, 197.11.10.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{197}$ |
Relative Eisenstein polynomial: | \( x^{22} + 394 \) |
Ramification polygon
Residual polynomials: | $z^{21} + 22z^{20} + 34z^{19} + 161z^{18} + 26z^{17} + 133z^{16} + 147z^{15} + 139z^{14} + 39z^{13} + 192z^{12} + 92z^{11} + 172z^{10} + 92z^{9} + 192z^{8} + 39z^{7} + 139z^{6} + 147z^{5} + 133z^{4} + 26z^{3} + 161z^{2} + 34z + 22$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $D_{22}$ (as 22T3) |
Inertia group: | $C_{22}$ (as 22T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $22$ |
Wild slopes: | None |
Galois mean slope: | $21/22$ |
Galois splitting model: | Not computed |