Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 187\cdot 191 + 86\cdot 191^{2} + 75\cdot 191^{3} + 94\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 116\cdot 191 + 107\cdot 191^{2} + 108\cdot 191^{3} + 72\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 62 + 169\cdot 191 + 71\cdot 191^{2} + 13\cdot 191^{3} + 156\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 + 159\cdot 191 + 106\cdot 191^{2} + 43\cdot 191^{3} + 63\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 141 + 86\cdot 191 + 74\cdot 191^{2} + 185\cdot 191^{3} + 99\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 148 + 19\cdot 191 + 93\cdot 191^{2} + 44\cdot 191^{3} + 146\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 154 + 9\cdot 191 + 128\cdot 191^{2} + 74\cdot 191^{3} + 53\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 159 + 15\cdot 191 + 95\cdot 191^{2} + 27\cdot 191^{3} + 78\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,7)(5,8)$ |
| $(1,3,8,7)(2,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,5)(3,7)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,3,8,7)(2,4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
