Defining polynomial
\(x^{14} + 49 x^{13} + 1043 x^{12} + 12593 x^{11} + 94409 x^{10} + 451927 x^{9} + 1369193 x^{8} + 2531313 x^{7} + 2738925 x^{6} + 1818103 x^{5} + 871157 x^{4} + 978033 x^{3} + 3155803 x^{2} + 6978181 x + 6677920\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 11 }) }$: | $2$ |
This field is not Galois over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{2})$, 11.7.6.1 |
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: | \( x^{7} + 11 \) $\ \in\Q_{11}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{6} + 7z^{5} + 10z^{4} + 2z^{3} + 2z^{2} + 10z + 7$ |
Associated inertia: | $3$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_7:C_6$ (as 14T5) |
Inertia group: | Intransitive group isomorphic to $C_7$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $7$ |
Wild slopes: | None |
Galois mean slope: | $6/7$ |
Galois splitting model: | Not computed |