Defining polynomial
\(x^{10} - 12 x^{9} + 174 x^{8} - 640 x^{7} - 40 x^{6} + 11424 x^{5} - 7984 x^{4} - 79360 x^{3} + 84112 x^{2} + 955968 x + 1404384\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $5$ |
Discriminant exponent $c$: | $15$ |
Discriminant root field: | $\Q_{2}(\sqrt{-2\cdot 5})$ |
Root number: | $-i$ |
$\card{ \Gal(K/\Q_{ 2 }) }$: | $10$ |
This field is Galois and abelian over $\Q_{2}.$ | |
Visible slopes: | $[3]$ |
Intermediate fields
$\Q_{2}(\sqrt{-2\cdot 5})$, 2.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 2.5.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{5} + x^{2} + 1 \) |
Relative Eisenstein polynomial: | \( x^{2} + \left(4 t^{3} + 4 t^{2} + 4 t\right) x + 12 t^{4} + 4 t^{3} + 12 t^{2} + 12 t + 2 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
Galois group: | $C_{10}$ (as 10T1) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_2$ |
Unramified degree: | $5$ |
Tame degree: | $1$ |
Wild slopes: | $[3]$ |
Galois mean slope: | $3/2$ |
Galois splitting model: | $x^{10} - 22 x^{8} + 176 x^{6} - 616 x^{4} + 880 x^{2} - 352$ |