Defining polynomial
\(x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $5$ |
Discriminant exponent $c$: | $15$ |
Discriminant root field: | $\Q_{2}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 2 }) }$: | $10$ |
This field is Galois and abelian over $\Q_{2}.$ | |
Visible slopes: | $[3]$ |
Intermediate fields
$\Q_{2}(\sqrt{2})$, 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: | \( x^{2} + 14 \) $\ \in\Q_{2}(t)[x]$ |
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: | $x^{10} - 18 x^{8} + 112 x^{6} - 280 x^{4} + 240 x^{2} - 32$ |