Defining polynomial
\(x^{14} + 4 x^{13} + 4 x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 2 x^{8} + 6 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 4 x + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $14$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $27$ |
Discriminant root field: | $\Q_{2}(\sqrt{-2})$ |
Root number: | $i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[3]$ |
Intermediate fields
2.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: | \( x^{14} + 4 x^{13} + 4 x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 2 x^{8} + 6 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 4 x + 2 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{12} + z^{10} + z^{8} + z^{6} + z^{4} + z^{2} + 1$ |
Associated inertia: | $1$,$3$ |
Indices of inseparability: | $[14, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\wr C_7:C_3$ (as 14T44) |
Inertia group: | $C_2\wr C_7$ (as 14T29) |
Wild inertia group: | $C_2^7$ |
Unramified degree: | $3$ |
Tame degree: | $7$ |
Wild slopes: | $[18/7, 18/7, 18/7, 20/7, 20/7, 20/7, 3]$ |
Galois mean slope: | $649/224$ |
Galois splitting model: | $x^{14} + 14 x^{12} + 70 x^{10} + 140 x^{8} + 56 x^{6} - 112 x^{4} - 56 x^{2} + 8$ |