Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 39 + 111\cdot 191 + 175\cdot 191^{2} + 190\cdot 191^{3} + 41\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 54 + 71\cdot 191 + 122\cdot 191^{2} + 175\cdot 191^{3} + 108\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 100 + 63\cdot 191 + 99\cdot 191^{2} + 63\cdot 191^{3} + 190\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 114 + 149\cdot 191 + 64\cdot 191^{2} + 116\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 139 + 2\cdot 191 + 128\cdot 191^{2} + 60\cdot 191^{3} + 54\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 160 + 97\cdot 191 + 7\cdot 191^{2} + 97\cdot 191^{3} + 135\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 163 + 9\cdot 191 + 105\cdot 191^{2} + 4\cdot 191^{3} + 101\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 187 + 66\cdot 191 + 61\cdot 191^{2} + 171\cdot 191^{3} + 15\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,4,7,2,6,5,3)$ |
| $(1,5,2,4)(3,6,7,8)$ |
| $(1,2)(3,7)(4,5)(6,8)$ |
| $(1,3)(2,7)(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,4)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,4,2,5)(3,8,7,6)$ | $0$ |
| $2$ | $8$ | $(1,8,4,7,2,6,5,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,7,5,8,2,3,4,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
