Defining polynomial
\(x^{15} + 10 x^{13} + 45 x^{12} + 40 x^{11} + 393 x^{10} + 890 x^{9} + 750 x^{8} - 3970 x^{7} + 9610 x^{6} + 13085 x^{5} + 151045 x^{4} + 26525 x^{3} + 116170 x^{2} - 53795 x + 67662\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{11}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 11 }) }$: | $15$ |
This field is Galois and abelian over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
11.3.0.1, 11.5.4.4 |
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: | \( x^{5} + 11 \) $\ \in\Q_{11}(t)[x]$ |
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: | $C_{15}$ (as 15T1) |
Inertia group: | Intransitive group isomorphic to $C_5$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $3$ |
Tame degree: | $5$ |
Wild slopes: | None |
Galois mean slope: | $4/5$ |
Galois splitting model: | Not computed |