Defining polynomial
\(x^{12} + 152\) |
Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $12$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $11$ |
Discriminant root field: | $\Q_{19}(\sqrt{19\cdot 2})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 19 }) }$: | $6$ |
This field is not Galois over $\Q_{19}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{19}(\sqrt{19})$, 19.3.2.2, 19.4.3.2, 19.6.5.2 |
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: | \( x^{12} + 152 \) |
Ramification polygon
Residual polynomials: | $z^{11} + 12z^{10} + 9z^{9} + 11z^{8} + z^{7} + 13z^{6} + 12z^{5} + 13z^{4} + z^{3} + 11z^{2} + 9z + 12$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_3\times D_4$ (as 12T14) |
Inertia group: | $C_{12}$ (as 12T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $12$ |
Wild slopes: | None |
Galois mean slope: | $11/12$ |
Galois splitting model: | Not computed |