Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 + 124\cdot 191 + 33\cdot 191^{2} + 120\cdot 191^{3} + 152\cdot 191^{4} + 25\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 + 111\cdot 191 + 110\cdot 191^{2} + 90\cdot 191^{3} + 136\cdot 191^{4} + 73\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 65 + 164\cdot 191 + 127\cdot 191^{2} + 153\cdot 191^{3} + 100\cdot 191^{4} + 4\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 77 + 173\cdot 191 + 55\cdot 191^{2} + 20\cdot 191^{3} + 43\cdot 191^{4} + 7\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 116 + 161\cdot 191 + 10\cdot 191^{3} + 82\cdot 191^{4} + 146\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 129 + 64\cdot 191 + 164\cdot 191^{2} + 47\cdot 191^{3} + 23\cdot 191^{4} + 90\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 144 + 3\cdot 191 + 56\cdot 191^{2} + 139\cdot 191^{3} + 116\cdot 191^{4} + 117\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 166 + 151\cdot 191 + 23\cdot 191^{2} + 182\cdot 191^{3} + 108\cdot 191^{4} + 107\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,4)(6,7)$ |
| $(1,3,7,2,8,4,6,5)$ |
| $(2,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,5)(3,4)(6,7)$ | $-4$ |
| $2$ | $2$ | $(2,5)(3,4)$ | $0$ |
| $4$ | $2$ | $(2,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,8,6)(2,4,5,3)$ | $0$ |
| $2$ | $4$ | $(1,7,8,6)(2,3,5,4)$ | $0$ |
| $4$ | $8$ | $(1,3,7,2,8,4,6,5)$ | $0$ |
| $4$ | $8$ | $(1,2,6,3,8,5,7,4)$ | $0$ |
| $4$ | $8$ | $(1,3,7,5,8,4,6,2)$ | $0$ |
| $4$ | $8$ | $(1,5,6,3,8,2,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
