Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 22\cdot 191 + 117\cdot 191^{2} + 125\cdot 191^{3} + 136\cdot 191^{4} + 83\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 40\cdot 191 + 171\cdot 191^{2} + 106\cdot 191^{3} + 116\cdot 191^{4} + 158\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 26\cdot 191 + 11\cdot 191^{2} + 96\cdot 191^{3} + 144\cdot 191^{4} + 104\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 + 114\cdot 191 + 45\cdot 191^{2} + 159\cdot 191^{3} + 13\cdot 191^{4} + 77\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 123 + 76\cdot 191 + 145\cdot 191^{2} + 31\cdot 191^{3} + 177\cdot 191^{4} + 113\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 154 + 164\cdot 191 + 179\cdot 191^{2} + 94\cdot 191^{3} + 46\cdot 191^{4} + 86\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 161 + 150\cdot 191 + 19\cdot 191^{2} + 84\cdot 191^{3} + 74\cdot 191^{4} + 32\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 178 + 168\cdot 191 + 73\cdot 191^{2} + 65\cdot 191^{3} + 54\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 |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7)(2,8)(4,5)$ |
| $(1,6,7,4,8,3,2,5)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,7)(2,8)(4,5)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,7,4,8,3,2,5)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,4,2,6,8,5,7,3)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
