Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 31\cdot 191 + 80\cdot 191^{2} + 186\cdot 191^{3} + 157\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 + 161\cdot 191 + 51\cdot 191^{2} + 100\cdot 191^{3} + 32\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 105 + 141\cdot 191 + 184\cdot 191^{2} + 110\cdot 191^{3} + 15\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 115 + 189\cdot 191 + 152\cdot 191^{2} + 83\cdot 191^{3} + 116\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 156 + 7\cdot 191 + 134\cdot 191^{2} + 2\cdot 191^{3} + 80\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 159 + 83\cdot 191 + 129\cdot 191^{2} + 3\cdot 191^{3} + 168\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 180 + 82\cdot 191 + 88\cdot 191^{2} + 16\cdot 191^{3} + 121\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 183 + 65\cdot 191 + 133\cdot 191^{2} + 68\cdot 191^{3} + 72\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,8,4,5,6,2,3)$ |
| $(1,5)(2,8)(3,4)(6,7)$ |
| $(1,8,5,2)(3,7,4,6)$ |
| $(1,8)(2,5)(6,7)$ |
| $(1,5)(2,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $-4$ |
| $2$ | $2$ | $(1,5)(2,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,8,5,2)(3,7,4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,5,8)(3,7,4,6)$ | $0$ |
| $4$ | $4$ | $(1,6,5,7)(2,3,8,4)$ | $0$ |
| $4$ | $8$ | $(1,7,8,4,5,6,2,3)$ | $0$ |
| $4$ | $8$ | $(1,7,2,3,5,6,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
