Defining polynomial
\(x^{12} + 225 x^{10} + 98 x^{9} + 17904 x^{8} + 10824 x^{7} + 611647 x^{6} + 498390 x^{5} + 8494833 x^{4} + 11764900 x^{3} + 38205036 x^{2} + 73669974 x + 36476587\) |
Invariants
Base field: | $\Q_{17}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 17 }) }$: | $12$ |
This field is Galois over $\Q_{17}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{17}(\sqrt{3})$, 17.3.2.1 x3, 17.4.0.1, 17.6.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 17.4.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{4} + 7 x^{2} + 10 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{3} + 51 x + 17 \) $\ \in\Q_{17}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_3:C_4$ (as 12T5) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | Not computed |