Defining polynomial
\(x^{12} - 704 x^{6} - 121\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 11 }) }$: | $6$ |
This field is not Galois over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{2})$, 11.4.2.2, 11.6.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} + 7 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{6} + 110 t + 33 \) $\ \in\Q_{11}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{5} + 6z^{4} + 4z^{3} + 9z^{2} + 4z + 6$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_3:C_{12}$ (as 12T19) |
Inertia group: | Intransitive group isomorphic to $C_6$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $6$ |
Wild slopes: | None |
Galois mean slope: | $5/6$ |
Galois splitting model: | $x^{12} - 3 x^{11} + 41 x^{10} - 273 x^{9} + 5320 x^{8} - 1216 x^{7} + 185714 x^{6} - 462051 x^{5} + 6328363 x^{4} + 19619049 x^{3} + 296778921 x^{2} + 695676814 x + 2813125384$ |