Defining polynomial
\(x^{10} + 60 x^{9} + 1450 x^{8} + 17760 x^{7} + 112360 x^{6} + 319418 x^{5} + 225500 x^{4} + 89240 x^{3} + 226880 x^{2} + 1274440 x + 3013497\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 13 }) }$: | $2$ |
This field is not Galois over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{13}(\sqrt{2})$, 13.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{13}(\sqrt{2})$ $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{2} + 12 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{5} + 13 \) $\ \in\Q_{13}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{4} + 5z^{3} + 10z^{2} + 10z + 5$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $F_5$ (as 10T4) |
Inertia group: | Intransitive group isomorphic to $C_5$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $5$ |
Wild slopes: | None |
Galois mean slope: | $4/5$ |
Galois splitting model: | $x^{10} - x^{8} - 26 x^{7} - 36 x^{6} - 52 x^{5} + 54 x^{4} + 130 x^{3} - 125 x^{2} - 125$ |