Defining polynomial
| \( x^{10} - 2 x^{9} + 2 x^{5} - 2 x^{4} + 2 x^{2} - 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $10$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $14$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-*})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.5.4.1 |
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: | \( x^{10} - 10 x^{9} - 1120 x^{8} - 4780 x^{7} + 224840 x^{6} + 1086138 x^{5} - 22976430 x^{4} - 20509260 x^{3} + 65579870 x^{2} - 274635860 x - 649182262 \) |
Invariants of the Galois closure
| Galois group: | $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) |
| Inertia group: | $C_2 \times (C_2^4 : C_5)$ |
| Unramified degree: | $4$ |
| Tame degree: | $5$ |
| Wild slopes: | [8/5, 8/5, 8/5, 8/5, 2] |
| Galois mean slope: | $71/40$ |
| Galois splitting model: | $x^{10} + x^{8} + 10 x^{6} - 10 x^{4} + 5 x^{2} + 5$ |