Defining polynomial
\(x^{6} + 53\) |
Invariants
Base field: | $\Q_{53}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $5$ |
Discriminant root field: | $\Q_{53}(\sqrt{53})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 53 }) }$: | $2$ |
This field is not Galois over $\Q_{53}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{53}(\sqrt{53})$, 53.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{53}$ |
Relative Eisenstein polynomial: | \( x^{6} + 53 \) |
Ramification polygon
Residual polynomials: | $z^{5} + 6z^{4} + 15z^{3} + 20z^{2} + 15z + 6$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $D_6$ (as 6T3) |
Inertia group: | $C_6$ (as 6T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $6$ |
Wild slopes: | None |
Galois mean slope: | $5/6$ |
Galois splitting model: | $x^{6} - 12 x^{5} + 60 x^{4} - 160 x^{3} + 240 x^{2} - 192 x + 11$ |