Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 62\cdot 191 + 145\cdot 191^{2} + 36\cdot 191^{3} + 88\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 98\cdot 191 + 170\cdot 191^{2} + 167\cdot 191^{3} + 190\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 125\cdot 191 + 134\cdot 191^{2} + 97\cdot 191^{3} + 71\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 65\cdot 191 + 165\cdot 191^{2} + 20\cdot 191^{3} + 42\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 70 + 155\cdot 191 + 172\cdot 191^{2} + 50\cdot 191^{3} + 185\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 86 + 83\cdot 191 + 182\cdot 191^{2} + 166\cdot 191^{3} + 143\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 143 + 183\cdot 191 + 7\cdot 191^{2} + 26\cdot 191^{3} + 155\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 188 + 181\cdot 191 + 166\cdot 191^{2} + 5\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 |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,5,8,6)(2,4,7,3)$ |
| $(3,6)(4,5)$ |
| $(1,8)(2,7)(3,4)(5,6)$ |
| $(1,5,2,4)(3,7,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $-4$ |
| $2$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,4)(3,8)(6,7)$ | $0$ |
| $4$ | $4$ | $(1,5,2,4)(3,7,6,8)$ | $0$ |
| $4$ | $4$ | $(1,6,8,5)(2,3,7,4)$ | $0$ |
| $4$ | $4$ | $(1,5,8,6)(2,4,7,3)$ | $0$ |
| $4$ | $4$ | $(3,5,6,4)(7,8)$ | $0$ |
| $4$ | $4$ | $(3,4,6,5)(7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
