Defining polynomial
\(x^{11} + 110 x^{10} + 495\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $11$ |
Ramification exponent $e$: | $11$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $20$ |
Discriminant root field: | $\Q_{11}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 11 }) }$: | $11$ |
This field is Galois and abelian over $\Q_{11}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{11}$ |
Relative Eisenstein polynomial: | \( x^{11} + 110 x^{10} + 495 \) |
Ramification polygon
Residual polynomials: | $z^{10} + 10$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[10, 0]$ |
Invariants of the Galois closure
Galois group: | $C_{11}$ (as 11T1) |
Inertia group: | $C_{11}$ (as 11T1) |
Wild inertia group: | $C_{11}$ |
Unramified degree: | $1$ |
Tame degree: | $1$ |
Wild slopes: | $[2]$ |
Galois mean slope: | $20/11$ |
Galois splitting model: | Not computed |