Defining polynomial
\(x^{14} + 2 x^{10} + 2 x^{9} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $14$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $18$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[12/7]$ |
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: | \( x^{14} + 2 x^{10} + 2 x^{9} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2 \) |
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: | $[5, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\wr C_7:C_3$ (as 14T44) |
Inertia group: | $C_2^3:F_8$ (as 14T21) |
Wild inertia group: | $C_2^6$ |
Unramified degree: | $6$ |
Tame degree: | $7$ |
Wild slopes: | $[8/7, 8/7, 8/7, 12/7, 12/7, 12/7]$ |
Galois mean slope: | $367/224$ |
Galois splitting model: | $x^{14} - 14 x^{12} - 14 x^{11} + 14 x^{10} + 140 x^{9} + 378 x^{8} - 236 x^{7} - 770 x^{6} - 1918 x^{5} - 1568 x^{4} + 4032 x^{3} + 3878 x^{2} + 7476 x - 470$ |