Defining polynomial
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$( x^{2} + 190 x + 19 )^{6} + 191$
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Invariants
| Base field: | $\Q_{191}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{191}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{191})$ $=$$\Gal(K/\Q_{191})$: | $D_6$ |
| This field is Galois over $\Q_{191}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $36480 = (191^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{191}(\sqrt{7})$, $\Q_{191}(\sqrt{191})$, $\Q_{191}(\sqrt{191\cdot 7})$, 191.1.3.2a1.1 x3, 191.2.2.2a1.2, 191.2.3.4a1.2, 191.1.6.5a1.2 x3, 191.1.6.5a1.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{191}(\sqrt{7})$ $\cong \Q_{191}(t)$ where $t$ is a root of
\( x^{2} + 190 x + 19 \)
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| Relative Eisenstein polynomial: |
\( x^{6} + 191 \)
$\ \in\Q_{191}(t)[x]$
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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: | $D_6$ (as 12T3) |
| 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: |
$x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 90 x^{8} - 126 x^{7} - 241 x^{6} + 1020 x^{5} + 2955 x^{4} - 7690 x^{3} + 2886 x^{2} + 1140 x + 36100$
|