Defining polynomial
\(x^{12} + 24 x^{10} + 74 x^{9} + 198 x^{8} + 480 x^{7} - 346 x^{6} + 5208 x^{5} - 3276 x^{4} - 856 x^{3} + 53628 x^{2} - 74328 x + 152743\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 11 }) }$: | $12$ |
This field is Galois over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{2})$, 11.3.2.1 x3, 11.4.0.1, 11.6.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 11.4.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{4} + 8 x^{2} + 10 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{3} + 11 \) $\ \in\Q_{11}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |