Defining polynomial
|
\(x^{12} + 358 x^{10} + 36 x^{9} + 53003 x^{8} + 72 x^{7} + 4118916 x^{6} - 737784 x^{5} + 177619595 x^{4} - 58235760 x^{3} + 4088219302 x^{2} - 1289138436 x + 40329774445\)
|
Invariants
| Base field: | $\Q_{59}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{59}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 59 }) }$: | $12$ |
| This field is Galois and abelian over $\Q_{59}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{59}(\sqrt{2})$, $\Q_{59}(\sqrt{59})$, $\Q_{59}(\sqrt{59\cdot 2})$, 59.3.0.1, 59.4.2.1, 59.6.0.1, 59.6.3.1, 59.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 59.6.0.1 $\cong \Q_{59}(t)$ where $t$ is a root of
\( x^{6} + 2 x^{4} + 18 x^{3} + 38 x^{2} + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 59 \)
$\ \in\Q_{59}(t)[x]$
|
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 |