Defining polynomial
|
\(x^{12} + 918 x^{11} + 351241 x^{10} + 71712630 x^{9} + 8244584136 x^{8} + 506874732756 x^{7} + 13125344775560 x^{6} + 9625198256031 x^{5} + 28457943732288 x^{4} + 16844354225613 x^{3} + 132306217741765 x^{2} + 68598705820311 x + 44162739951115\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{17}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 17 }) }$: | $12$ |
| This field is Galois and abelian over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{17}(\sqrt{3})$, $\Q_{17}(\sqrt{17})$, $\Q_{17}(\sqrt{17\cdot 3})$, 17.3.0.1, 17.4.2.1, 17.6.0.1, 17.6.3.1, 17.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 17.6.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of
\( x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 153 x + 17 \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_6$ (as 12T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |