Defining polynomial
\(x^{4} - 85305 x^{3} - 42072520 x^{2} - 585385 x + 11045\) |
Invariants
Base field: | $\Q_{47}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{47}(\sqrt{5})$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 47 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{47}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{47}(\sqrt{5})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{47}(\sqrt{5})$ $\cong \Q_{47}(t)$ where $t$ is a root of \( x^{2} + 45 x + 5 \) |
Relative Eisenstein polynomial: | \( x^{2} + \left(1927 t + 705\right) x + 47 t \) $\ \in\Q_{47}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_4$ (as 4T1) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | $x^{4} - x^{3} - 59 x^{2} + 59 x + 661$ |