Defining polynomial
\(x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065\) |
Invariants
Base field: | $\Q_{137}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{137}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 137 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{137}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{137}(\sqrt{3})$, $\Q_{137}(\sqrt{137})$, $\Q_{137}(\sqrt{137\cdot 3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{137}(\sqrt{3})$ $\cong \Q_{137}(t)$ where $t$ is a root of \( x^{2} + 131 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{2} + 10138 x + 137 \) $\ \in\Q_{137}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_2^2$ (as 4T2) |
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: | $x^{4} + 1507 x^{2} + 675684$ |