Defining polynomial
| \( x^{12} - 11 x^{6} + 847 \) |
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $10$ |
| Discriminant root field: | $\Q_{11}(\sqrt{*})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 11 })|$: | $6$ |
| This field is not Galois over $\Q_{11}$. | |
Intermediate fields
| $\Q_{11}(\sqrt{*})$, 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: | $\Q_{11}(\sqrt{*})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} - x + 7 \) |
| Relative Eisenstein polynomial: | $ x^{6} - 11 t \in\Q_{11}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times C_3:C_4$ (as 12T19) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Unramified degree: | $6$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | $x^{12} - 3 x^{11} + 41 x^{10} - 273 x^{9} + 5320 x^{8} - 1216 x^{7} + 185714 x^{6} - 462051 x^{5} + 6328363 x^{4} + 19619049 x^{3} + 296778921 x^{2} + 695676814 x + 2813125384$ |