Defining polynomial
|
\(x^{17} + 136 x^{12} + 17\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $17$ |
| Ramification index $e$: | $17$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $28$ |
| Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{17})$: | $C_1$ |
| This field is not Galois over $\Q_{17}.$ | |
| Visible Artin slopes: | $[\frac{7}{4}]$ |
| Visible Swan slopes: | $[\frac{3}{4}]$ |
| Means: | $\langle\frac{12}{17}\rangle$ |
| Rams: | $(\frac{3}{4})$ |
| Jump set: | undefined |
| Roots of unity: | $16 = (17 - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 17 }$. |
Canonical tower
| Unramified subfield: | $\Q_{17}$ |
| Relative Eisenstein polynomial: |
\( x^{17} + 136 x^{12} + 17 \)
|
Ramification polygon
| Residual polynomials: | $z^4 + 6$ |
| Associated inertia: | $4$ |
| Indices of inseparability: | $[12, 0]$ |
Invariants of the Galois closure
| Galois degree: | $272$ |
| Galois group: | $F_{17}$ (as 17T5) |
| Inertia group: | $C_{17}:C_4$ (as 17T3) |
| Wild inertia group: | $C_{17}$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{7}{4}]$ |
| Galois Swan slopes: | $[\frac{3}{4}]$ |
| Galois mean slope: | $1.6911764705882353$ |
| Galois splitting model: | not computed |