Defining polynomial
\(x^{12} + 23\) |
Invariants
Base field: | $\Q_{23}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $12$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $11$ |
Discriminant root field: | $\Q_{23}(\sqrt{23})$ |
Root number: | $i$ |
$\card{ \Aut(K/\Q_{ 23 }) }$: | $2$ |
This field is not Galois over $\Q_{23}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{23}(\sqrt{23\cdot 5})$, 23.3.2.1, 23.4.3.1, 23.6.5.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{23}$ |
Relative Eisenstein polynomial: | \( x^{12} + 23 \) |
Ramification polygon
Residual polynomials: | $z^{11} + 12z^{10} + 20z^{9} + 13z^{8} + 12z^{7} + 10z^{6} + 4z^{5} + 10z^{4} + 12z^{3} + 13z^{2} + 20z + 12$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $D_{12}$ (as 12T12) |
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 |