Defining polynomial
|
\(x^{12} + 9 x^{10} + 88 x^{9} + 33 x^{8} + 216 x^{7} - 1299 x^{6} - 78 x^{5} - 1797 x^{4} - 15494 x^{3} + 21687 x^{2} - 41586 x + 201846\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 13 }) }$: | $12$ |
| This field is Galois and abelian over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{13}(\sqrt{2})$, 13.3.2.2, 13.4.0.1, 13.6.4.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 13.4.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{4} + 3 x^{2} + 12 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 13 \)
$\ \in\Q_{13}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + 3z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{12}$ (as 12T1) |
| 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: | $x^{12} - x^{11} + 5 x^{10} - 10 x^{9} + 31 x^{8} + 50 x^{7} + 84 x^{6} + 85 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1$ |