Defining polynomial
\(x^{10} - 165 x^{5} - 4356\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 11 }) }$: | $10$ |
This field is Galois and abelian over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{2})$, 11.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} + 7 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{5} + 33 t + 33 \) $\ \in\Q_{11}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{4} + 5z^{3} + 10z^{2} + 10z + 5$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{10}$ (as 10T1) |
Inertia group: | Intransitive group isomorphic to $C_5$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $5$ |
Wild slopes: | None |
Galois mean slope: | $4/5$ |
Galois splitting model: | $x^{10} - x^{9} + 2 x^{8} - 356 x^{7} - 877 x^{6} + 3793 x^{5} + 65887 x^{4} + 386316 x^{3} + 977865 x^{2} + 1230866 x + 2368673$ |