Defining polynomial
\( x^{14} + 2 x^{12} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{5} + 2 x^{2} + 2 \) |
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: | $1$ |
$|\Aut(K/\Q_{ 2 })|$: | $2$ |
This field is not Galois over $\Q_{2}.$ |
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: | \( x^{14} + 2 x^{12} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{5} + 2 x^{2} + 2 \) |
Invariants of the Galois closure
Galois group: | 14T44 |
Inertia group: | 14T21 |
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: | $x^{14} - 14 x^{12} - 14 x^{11} + 14 x^{10} + 140 x^{9} + 378 x^{8} - 236 x^{7} - 770 x^{6} - 1918 x^{5} - 1568 x^{4} + 4032 x^{3} + 3878 x^{2} + 7476 x - 470$ |