Defining polynomial
\(x^{11} + 33 x^{5} + 11\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $11$ |
Ramification exponent $e$: | $11$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $15$ |
Discriminant root field: | $\Q_{11}(\sqrt{11\cdot 2})$ |
Root number: | $i$ |
$\card{ \Aut(K/\Q_{ 11 }) }$: | $1$ |
This field is not Galois over $\Q_{11}.$ | |
Visible slopes: | $[3/2]$ |
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: | \( x^{11} + 33 x^{5} + 11 \) |
Ramification polygon
Residual polynomials: | $z^{5} + 7$ |
Associated inertia: | $5$ |
Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
Galois group: | $F_{11}$ (as 11T4) |
Inertia group: | $D_{11}$ (as 11T2) |
Wild inertia group: | $C_{11}$ |
Unramified degree: | $5$ |
Tame degree: | $2$ |
Wild slopes: | $[3/2]$ |
Galois mean slope: | $31/22$ |
Galois splitting model: | Not computed |