Defining polynomial
\(x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148\) |
Invariants
Base field: | $\Q_{67}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{67}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 67 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{67}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{67}(\sqrt{2})$, $\Q_{67}(\sqrt{67})$, $\Q_{67}(\sqrt{67\cdot 2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{67}(\sqrt{2})$ $\cong \Q_{67}(t)$ where $t$ is a root of \( x^{2} + 63 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{2} + 67 \) $\ \in\Q_{67}(t)[x]$ |
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: | $x^{4} + 1541 x^{2} + 646416$ |