Defining polynomial
|
\(x^{13} + 130 x^{3} + 13\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $13$ |
| Ramification exponent $e$: | $13$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{13}(\sqrt{13})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 13 }) }$: | $1$ |
| This field is not Galois over $\Q_{13}.$ | |
| Visible slopes: | $[5/4]$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 13 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{13}$ |
| Relative Eisenstein polynomial: |
\( x^{13} + 130 x^{3} + 13 \)
|
Ramification polygon
| Residual polynomials: | $z^{3} + 9$ |
| Associated inertia: | $3$ |
| Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
| Galois group: | $F_{13}$ (as 13T6) |
| Inertia group: | $C_{13}:C_4$ (as 13T4) |
| Wild inertia group: | $C_{13}$ |
| Unramified degree: | $3$ |
| Tame degree: | $4$ |
| Wild slopes: | $[5/4]$ |
| Galois mean slope: | $63/52$ |
| Galois splitting model: | Not computed |