Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 99\cdot 191 + 77\cdot 191^{2} + 125\cdot 191^{3} + 151\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 + 54\cdot 191 + 9\cdot 191^{2} + 173\cdot 191^{3} + 58\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 110 + 140\cdot 191 + 142\cdot 191^{2} + 168\cdot 191^{3} + 33\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 131 + 168\cdot 191 + 171\cdot 191^{2} + 143\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 144 + 36\cdot 191 + 110\cdot 191^{2} + 98\cdot 191^{3} + 49\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 149 + 175\cdot 191 + 105\cdot 191^{2} + 106\cdot 191^{3} + 67\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 150 + 185\cdot 191^{2} + 175\cdot 191^{3} + 121\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 184 + 87\cdot 191 + 152\cdot 191^{2} + 105\cdot 191^{3} + 137\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,2,7)(3,4,8,6)$ |
| $(3,8)(4,6)$ |
| $(1,2)(3,8)(4,6)(5,7)$ |
| $(1,6)(2,4)(3,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,6)(2,4)(3,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(3,8)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,5,2,7)(3,4,8,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,2,5)(3,6,8,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,2,8)(4,7,6,5)$ | $0$ |
| $2$ | $4$ | $(1,4,2,6)(3,5,8,7)$ | $0$ |
| $2$ | $4$ | $(1,5,2,7)(3,6,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
