Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 42\cdot 191 + 53\cdot 191^{2} + 148\cdot 191^{3} + 153\cdot 191^{4} + 110\cdot 191^{5} + 175\cdot 191^{6} + 54\cdot 191^{7} + 120\cdot 191^{8} + 47\cdot 191^{9} + 2\cdot 191^{10} +O\left(191^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 97\cdot 191 + 22\cdot 191^{2} + 160\cdot 191^{3} + 150\cdot 191^{4} + 172\cdot 191^{5} + 174\cdot 191^{6} + 42\cdot 191^{7} + 145\cdot 191^{8} + 125\cdot 191^{9} + 174\cdot 191^{10} +O\left(191^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 42 + 172\cdot 191 + 141\cdot 191^{2} + 40\cdot 191^{3} + 5\cdot 191^{4} + 97\cdot 191^{5} + 170\cdot 191^{6} + 133\cdot 191^{7} + 91\cdot 191^{8} + 141\cdot 191^{9} + 12\cdot 191^{10} +O\left(191^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 84 + 29\cdot 191 + 31\cdot 191^{2} + 53\cdot 191^{3} + 126\cdot 191^{4} + 32\cdot 191^{5} + 152\cdot 191^{6} + 155\cdot 191^{7} + 168\cdot 191^{8} + 184\cdot 191^{9} + 87\cdot 191^{10} +O\left(191^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 107 + 161\cdot 191 + 159\cdot 191^{2} + 137\cdot 191^{3} + 64\cdot 191^{4} + 158\cdot 191^{5} + 38\cdot 191^{6} + 35\cdot 191^{7} + 22\cdot 191^{8} + 6\cdot 191^{9} + 103\cdot 191^{10} +O\left(191^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 149 + 18\cdot 191 + 49\cdot 191^{2} + 150\cdot 191^{3} + 185\cdot 191^{4} + 93\cdot 191^{5} + 20\cdot 191^{6} + 57\cdot 191^{7} + 99\cdot 191^{8} + 49\cdot 191^{9} + 178\cdot 191^{10} +O\left(191^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 171 + 93\cdot 191 + 168\cdot 191^{2} + 30\cdot 191^{3} + 40\cdot 191^{4} + 18\cdot 191^{5} + 16\cdot 191^{6} + 148\cdot 191^{7} + 45\cdot 191^{8} + 65\cdot 191^{9} + 16\cdot 191^{10} +O\left(191^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 179 + 148\cdot 191 + 137\cdot 191^{2} + 42\cdot 191^{3} + 37\cdot 191^{4} + 80\cdot 191^{5} + 15\cdot 191^{6} + 136\cdot 191^{7} + 70\cdot 191^{8} + 143\cdot 191^{9} + 188\cdot 191^{10} +O\left(191^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6,4,7,8,3,5,2)$ |
| $(1,7,8,2)(3,4,6,5)$ |
| $(1,5,8,4)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,4)(3,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $4$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,6,4,7,8,3,5,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,4,2,8,6,5,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
