Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 63\cdot 191 + 8\cdot 191^{2} + 178\cdot 191^{3} + 36\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 91\cdot 191 + 72\cdot 191^{2} + 105\cdot 191^{3} + 104\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 + 150\cdot 191 + 166\cdot 191^{2} + 107\cdot 191^{3} + 52\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 74 + 76\cdot 191 + 122\cdot 191^{2} + 89\cdot 191^{3} + 66\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 96 + 15\cdot 191 + 182\cdot 191^{2} + 166\cdot 191^{3} + 163\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 138 + 190\cdot 191 + 93\cdot 191^{2} + 108\cdot 191^{3} + 150\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 178 + 136\cdot 191 + 87\cdot 191^{2} + 100\cdot 191^{3} + 172\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 182 + 39\cdot 191 + 30\cdot 191^{2} + 98\cdot 191^{3} + 16\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,5)(6,7)$ |
| $(1,3)(2,8)(4,6)(5,7)$ |
| $(1,3)(4,6)$ |
| $(1,6,3,4)(2,5,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(4,6)$ | $0$ |
| $1$ | $4$ | $(1,4,3,6)(2,5,8,7)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,3,4)(2,7,8,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,3,4)(2,5,8,7)$ | $0$ |
| $2$ | $4$ | $(1,2,3,8)(4,5,6,7)$ | $0$ |
| $2$ | $4$ | $(1,5,3,7)(2,4,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
