Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 42 + 124\cdot 191 + 79\cdot 191^{2} + 164\cdot 191^{3} + 49\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 56 + 8\cdot 191 + 97\cdot 191^{2} + 140\cdot 191^{3} + 10\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 62 + 42\cdot 191 + 88\cdot 191^{2} + 177\cdot 191^{3} + 101\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 76 + 117\cdot 191 + 105\cdot 191^{2} + 153\cdot 191^{3} + 62\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 115 + 73\cdot 191 + 85\cdot 191^{2} + 37\cdot 191^{3} + 128\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 129 + 148\cdot 191 + 102\cdot 191^{2} + 13\cdot 191^{3} + 89\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 135 + 182\cdot 191 + 93\cdot 191^{2} + 50\cdot 191^{3} + 180\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 149 + 66\cdot 191 + 111\cdot 191^{2} + 26\cdot 191^{3} + 141\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,3,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
