Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 + 129\cdot 191 + 143\cdot 191^{2} + 136\cdot 191^{3} + 94\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 + 189\cdot 191 + 51\cdot 191^{2} + 4\cdot 191^{3} + 74\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 + 24\cdot 191 + 34\cdot 191^{2} + 174\cdot 191^{3} + 155\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 61 + 125\cdot 191 + 147\cdot 191^{2} + 154\cdot 191^{3} + 58\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 + 124\cdot 191 + 119\cdot 191^{2} + 163\cdot 191^{3} + 68\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 82 + 134\cdot 191 + 59\cdot 191^{2} + 27\cdot 191^{3} + 125\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 100 + 96\cdot 191 + 149\cdot 191^{2} + 166\cdot 191^{3} + 8\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 144 + 131\cdot 191 + 57\cdot 191^{2} + 127\cdot 191^{3} + 177\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,3,2)(4,8,5,6)$ |
| $(1,7,8,4,3,2,6,5)$ |
| $(2,7)(4,5)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,8,3,6)(2,4,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$-4$ |
| $2$ |
$2$ |
$(2,7)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,6)(3,5)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(2,5)(4,7)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(3,6)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,3,6)(2,4,7,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,3,8)(2,4,7,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,3,2)(4,8,5,6)$ |
$0$ |
| $4$ |
$8$ |
$(1,5,6,2,3,4,8,7)$ |
$0$ |
| $4$ |
$8$ |
$(1,5,8,7,3,4,6,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
