Defining polynomial
\(x^{13} + 5\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $13$ |
Ramification exponent $e$: | $13$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}$ |
Relative Eisenstein polynomial: | \( x^{13} + 5 \) |
Ramification polygon
Residual polynomials: | $z^{12} + 3z^{11} + 3z^{10} + z^{9} + 2z^{7} + z^{6} + z^{5} + 2z^{4} + z^{2} + 3z + 3$ |
Associated inertia: | $4$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{13}:C_4$ (as 13T4) |
Inertia group: | $C_{13}$ (as 13T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $13$ |
Wild slopes: | None |
Galois mean slope: | $12/13$ |
Galois splitting model: | Not computed |