Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 52\cdot 191 + 59\cdot 191^{2} + 187\cdot 191^{3} + 107\cdot 191^{4} + 74\cdot 191^{5} + 114\cdot 191^{6} + 40\cdot 191^{7} +O\left(191^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 158\cdot 191 + 6\cdot 191^{2} + 20\cdot 191^{3} + 91\cdot 191^{4} + 32\cdot 191^{5} + 15\cdot 191^{6} + 131\cdot 191^{7} +O\left(191^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 76 + 7\cdot 191 + 108\cdot 191^{2} + 113\cdot 191^{3} + 82\cdot 191^{4} + 121\cdot 191^{5} + 142\cdot 191^{6} + 12\cdot 191^{7} +O\left(191^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 93 + 191 + 21\cdot 191^{2} + 36\cdot 191^{3} + 40\cdot 191^{4} + 171\cdot 191^{5} + 106\cdot 191^{6} + 137\cdot 191^{7} +O\left(191^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 96 + 82\cdot 191 + 84\cdot 191^{2} + 51\cdot 191^{3} + 34\cdot 191^{4} + 67\cdot 191^{5} + 114\cdot 191^{6} + 65\cdot 191^{7} +O\left(191^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 140 + 162\cdot 191 + 125\cdot 191^{2} + 24\cdot 191^{3} + 164\cdot 191^{4} + 141\cdot 191^{5} + 185\cdot 191^{6} + 107\cdot 191^{7} +O\left(191^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 172 + 51\cdot 191 + 127\cdot 191^{2} + 39\cdot 191^{3} + 168\cdot 191^{4} + 166\cdot 191^{5} + 12\cdot 191^{6} + 138\cdot 191^{7} +O\left(191^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 176 + 56\cdot 191 + 40\cdot 191^{2} + 100\cdot 191^{3} + 75\cdot 191^{4} + 179\cdot 191^{5} + 71\cdot 191^{6} + 130\cdot 191^{7} +O\left(191^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,5,4)(2,8)(6,7)$ |
| $(1,5)(2,7)$ |
| $(2,7)(6,8)$ |
| $(1,3,2,8)(4,7,6,5)$ |
| $(3,4)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,5)(2,7)(3,4)(6,8)$ |
$-4$ |
| $2$ |
$2$ |
$(3,4)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(3,4)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,8)(4,6)(5,7)$ |
$0$ |
| $8$ |
$2$ |
$(1,5)(3,8)(4,6)$ |
$0$ |
| $4$ |
$4$ |
$(3,6,4,8)$ |
$2$ |
| $4$ |
$4$ |
$(1,5)(2,7)(3,8,4,6)$ |
$-2$ |
| $4$ |
$4$ |
$(1,7,5,2)(3,6,4,8)$ |
$0$ |
| $8$ |
$4$ |
$(1,3,5,4)(2,8)(6,7)$ |
$0$ |
| $8$ |
$4$ |
$(1,4,5,3)(2,8)(6,7)$ |
$0$ |
| $8$ |
$4$ |
$(1,3,2,8)(4,7,6,5)$ |
$0$ |
| $8$ |
$4$ |
$(1,8,2,3)(4,5,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
