Defining polynomial
\(x^{13} + 53\) |
Invariants
Base field: | $\Q_{53}$ |
Degree $d$: | $13$ |
Ramification exponent $e$: | $13$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{53}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 53 }) }$: | $13$ |
This field is Galois and abelian over $\Q_{53}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 53 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{53}$ |
Relative Eisenstein polynomial: | \( x^{13} + 53 \) |
Ramification polygon
Residual polynomials: | $z^{12} + 13z^{11} + 25z^{10} + 21z^{9} + 26z^{8} + 15z^{7} + 20z^{6} + 20z^{5} + 15z^{4} + 26z^{3} + 21z^{2} + 25z + 13$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{13}$ (as 13T1) |
Inertia group: | $C_{13}$ (as 13T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $1$ |
Tame degree: | $13$ |
Wild slopes: | None |
Galois mean slope: | $12/13$ |
Galois splitting model: | Not computed |