Defining polynomial
|
\(x^{12} + 144 x^{11} + 8652 x^{10} + 277920 x^{9} + 5045820 x^{8} + 49440384 x^{7} + 211217114 x^{6} + 98884944 x^{5} + 20432100 x^{4} + 10157760 x^{3} + 142459992 x^{2} + 1361530944 x + 5427130041\)
|
Invariants
| Base field: | $\Q_{29}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{29}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 29 }) }$: | $12$ |
| This field is Galois over $\Q_{29}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{29}(\sqrt{2})$, $\Q_{29}(\sqrt{29})$, $\Q_{29}(\sqrt{29\cdot 2})$, 29.3.2.1 x3, 29.4.2.1, 29.6.4.1, 29.6.5.1 x3, 29.6.5.2 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{29}(\sqrt{2})$ $\cong \Q_{29}(t)$ where $t$ is a root of
\( x^{2} + 24 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + 29 \)
$\ \in\Q_{29}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{5} + 6z^{4} + 15z^{3} + 20z^{2} + 15z + 6$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_6$ (as 12T3) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | Not computed |