Defining polynomial
\(x^{12} + 8 x^{10} + 36 x^{9} + 57 x^{8} + 216 x^{7} + 342 x^{6} - 4320 x^{5} + 2675 x^{4} + 2412 x^{3} + 26122 x^{2} + 17316 x + 16164\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $9$ |
Discriminant root field: | $\Q_{11}(\sqrt{11})$ |
Root number: | $i$ |
$\card{ \Aut(K/\Q_{ 11 }) }$: | $6$ |
This field is not Galois over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{11\cdot 2})$, 11.3.0.1, 11.4.3.1, 11.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 11.3.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{3} + 2 x + 9 \) |
Relative Eisenstein polynomial: | \( x^{4} + 11 \) $\ \in\Q_{11}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_3\times D_4$ (as 12T14) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | $x^{12} - x^{11} - 7 x^{10} + 10 x^{9} + 77 x^{8} - 8 x^{7} - 171 x^{6} + 22 x^{5} + 476 x^{4} - 192 x^{3} - 112 x^{2} + 64 x + 64$ |