Defining polynomial
|
\(x^{13} + 13 x^{4} + 13\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $13$ |
| Ramification index $e$: | $13$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{13}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{13})$: | $C_1$ |
| This field is not Galois over $\Q_{13}.$ | |
| Visible Artin slopes: | $[\frac{4}{3}]$ |
| Visible Swan slopes: | $[\frac{1}{3}]$ |
| Means: | $\langle\frac{4}{13}\rangle$ |
| Rams: | $(\frac{1}{3})$ |
| Jump set: | undefined |
| Roots of unity: | $12 = (13 - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 13 }$. |
Canonical tower
| Unramified subfield: | $\Q_{13}$ |
| Relative Eisenstein polynomial: |
\( x^{13} + 13 x^{4} + 13 \)
|
Ramification polygon
| Residual polynomials: | $z^4 + 9$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $78$ |
| Galois group: | $C_{13}:C_6$ (as 13T5) |
| Inertia group: | $C_{13}:C_3$ (as 13T3) |
| Wild inertia group: | $C_{13}$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}]$ |
| Galois Swan slopes: | $[\frac{1}{3}]$ |
| Galois mean slope: | $1.2820512820512822$ |
| Galois splitting model: | not computed |