Defining polynomial
|
$( x^{2} + 177 x + 2 )^{6} + 724 x + 32399$
|
Invariants
| Base field: | $\Q_{181}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{181}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{181})$ $=$$\Gal(K/\Q_{181})$: | $C_2\times C_6$ |
| This field is Galois and abelian over $\Q_{181}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $32760 = (181^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{181}(\sqrt{2})$, $\Q_{181}(\sqrt{181})$, $\Q_{181}(\sqrt{181\cdot 2})$, 181.1.3.2a1.2, 181.2.2.2a1.2, 181.2.3.4a1.3, 181.1.6.5a1.5, 181.1.6.5a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{181}(\sqrt{2})$ $\cong \Q_{181}(t)$ where $t$ is a root of
\( x^{2} + 177 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + 724 t + 32399 \)
$\ \in\Q_{181}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^5 + 6 z^4 + 15 z^3 + 20 z^2 + 15 z + 6$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $12$ |
| Galois group: | $C_2\times C_6$ (as 12T2) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $6$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.8333333333333334$ |
| Galois splitting model: | not computed |