Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 94\cdot 191 + 36\cdot 191^{2} + 190\cdot 191^{3} + 73\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 67\cdot 191 + 145\cdot 191^{2} + 173\cdot 191^{3} + 61\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 62 + 57\cdot 191 + 79\cdot 191^{2} + 29\cdot 191^{3} + 124\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 105 + 169\cdot 191 + 174\cdot 191^{2} + 144\cdot 191^{3} + 16\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 107 + 78\cdot 191 + 13\cdot 191^{2} + 146\cdot 191^{3} + 125\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 109 + 76\cdot 191 + 114\cdot 191^{2} + 61\cdot 191^{3} + 115\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 167 + 75\cdot 191 + 91\cdot 191^{2} + 135\cdot 191^{3} + 120\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 174 + 144\cdot 191 + 108\cdot 191^{2} + 73\cdot 191^{3} + 125\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,8)(3,6)(4,5)$ |
| $(3,6)(4,5)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(2,8)(3,4,6,5)$ |
| $(1,4)(2,6)(3,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $0$ |
| $4$ | $4$ | $(1,6,8,4)(2,5,7,3)$ | $0$ |
| $4$ | $4$ | $(1,4,8,6)(2,3,7,5)$ | $0$ |
| $4$ | $4$ | $(1,3,7,6)(2,5,8,4)$ | $0$ |
| $4$ | $4$ | $(1,8,7,2)(3,6)$ | $0$ |
| $4$ | $4$ | $(1,2,7,8)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
