Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 59\cdot 191 + 183\cdot 191^{2} + 173\cdot 191^{3} + 137\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 75\cdot 191 + 91\cdot 191^{2} + 84\cdot 191^{3} + 50\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 88\cdot 191 + 68\cdot 191^{2} + 168\cdot 191^{3} + 145\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 + 93\cdot 191 + 9\cdot 191^{2} + 78\cdot 191^{3} + 32\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 + 189\cdot 191 + 112\cdot 191^{2} + 174\cdot 191^{3} + 49\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 62 + 94\cdot 191 + 38\cdot 191^{2} + 7\cdot 191^{3} + 5\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 166 + 174\cdot 191 + 8\cdot 191^{2} + 23\cdot 191^{3} + 16\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 181 + 180\cdot 191 + 59\cdot 191^{2} + 54\cdot 191^{3} + 135\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,4,8)(2,6,5,3)$ |
| $(1,4)(2,7,5,8)$ |
| $(2,5)(7,8)$ |
| $(1,4)(2,5)(3,6)(7,8)$ |
| $(1,3)(2,7)(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $-4$ |
| $2$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $0$ |
| $4$ | $4$ | $(1,7,4,8)(2,6,5,3)$ | $0$ |
| $4$ | $4$ | $(1,5,3,7)(2,6,8,4)$ | $0$ |
| $4$ | $4$ | $(1,7,3,5)(2,4,8,6)$ | $0$ |
| $4$ | $4$ | $(1,3,4,6)(2,5)$ | $0$ |
| $4$ | $4$ | $(1,6,4,3)(2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
