Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 34 + 60\cdot 191 + 115\cdot 191^{2} + 44\cdot 191^{3} + 179\cdot 191^{4} + 69\cdot 191^{5} + 86\cdot 191^{6} +O\left(191^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 68 + 34\cdot 191 + 153\cdot 191^{2} + 114\cdot 191^{4} + 23\cdot 191^{5} + 112\cdot 191^{6} +O\left(191^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 84 + 47\cdot 191 + 149\cdot 191^{2} + 99\cdot 191^{3} + 104\cdot 191^{4} + 5\cdot 191^{5} + 50\cdot 191^{6} +O\left(191^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 89 + 101\cdot 191 + 86\cdot 191^{2} + 27\cdot 191^{3} + 89\cdot 191^{4} + 93\cdot 191^{5} + 120\cdot 191^{6} +O\left(191^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 108 + 86\cdot 191 + 68\cdot 191^{2} + 129\cdot 191^{3} + 155\cdot 191^{4} + 132\cdot 191^{5} + 67\cdot 191^{6} +O\left(191^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 109 + 174\cdot 191 + 161\cdot 191^{2} + 83\cdot 191^{3} + 172\cdot 191^{4} + 39\cdot 191^{5} + 41\cdot 191^{6} +O\left(191^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 114 + 184\cdot 191 + 97\cdot 191^{2} + 44\cdot 191^{3} + 133\cdot 191^{4} + 13\cdot 191^{5} + 132\cdot 191^{6} +O\left(191^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 159 + 74\cdot 191 + 122\cdot 191^{2} + 142\cdot 191^{3} + 6\cdot 191^{4} + 3\cdot 191^{5} + 154\cdot 191^{6} +O\left(191^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,3)(7,8)$ |
| $(4,6)(7,8)$ |
| $(1,7,3,6,5,8,2,4)$ |
| $(1,2,5,3)(4,7,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,6)(7,8)$ |
$-4$ |
| $2$ |
$2$ |
$(4,6)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(2,3)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,5,3)(4,7,6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,5,2)(4,7,6,8)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,3,6,5,8,2,4)$ |
$0$ |
| $4$ |
$8$ |
$(1,6,2,7,5,4,3,8)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,2,6,5,8,3,4)$ |
$0$ |
| $4$ |
$8$ |
$(1,6,3,7,5,4,2,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
