Defining polynomial
\(x^{10} + 5 x + 10\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $10$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{5}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | $[9/8]$ |
Intermediate fields
$\Q_{5}(\sqrt{5\cdot 2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}$ |
Relative Eisenstein polynomial: | \( x^{10} + 5 x + 10 \) |
Ramification polygon
Residual polynomials: | $2z + 2$,$z^{5} + 2$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_5^2:\OD_{16}$ (as 10T28) |
Inertia group: | $C_5^2:C_8$ (as 10T18) |
Wild inertia group: | $C_5^2$ |
Unramified degree: | $2$ |
Tame degree: | $8$ |
Wild slopes: | $[9/8, 9/8]$ |
Galois mean slope: | $223/200$ |
Galois splitting model: | $x^{10} - 1955 x^{8} - 580 x^{7} - 2001110 x^{6} + 7057964 x^{5} - 243542310 x^{4} + 3251676820 x^{3} + 49946545205 x^{2} + 101082545540 x - 111230164631$ |