Defining polynomial
\(x^{12} + 11\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $12$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $11$ |
Discriminant root field: | $\Q_{11}(\sqrt{11})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 11 }) }$: | $2$ |
This field is not Galois over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{11\cdot 2})$, 11.3.2.1, 11.4.3.1, 11.6.5.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{11}$ |
Relative Eisenstein polynomial: | \( x^{12} + 11 \) |
Ramification polygon
Residual polynomials: | $z^{11} + z^{10} + 1$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |