Defining polynomial
| \( x^{4} + 1541 x^{2} + 646416 \) |
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: | $1$ |
| $|\Gal(K/\Q_{ 67 })|$: | $4$ |
| This field is Galois and abelian over $\Q_{67}$. | |
Intermediate fields
| $\Q_{67}(\sqrt{*})$, $\Q_{67}(\sqrt{67})$, $\Q_{67}(\sqrt{67*})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{67}(\sqrt{*})$ $\cong \Q_{67}(t)$ where $t$ is a root of \( x^{2} - x + 12 \) |
| Relative Eisenstein polynomial: | $ x^{2} - 67 t^{2} \in\Q_{67}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2^2$ (as 4T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{4} + 1541 x^{2} + 646416$ |