Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 36\cdot 191 + 75\cdot 191^{2} + 188\cdot 191^{3} + 2\cdot 191^{4} + 130\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 106\cdot 191 + 136\cdot 191^{2} + 165\cdot 191^{3} + 122\cdot 191^{4} + 97\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 146\cdot 191 + 90\cdot 191^{2} + 42\cdot 191^{3} + 73\cdot 191^{4} + 59\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 187\cdot 191 + 59\cdot 191^{2} + 53\cdot 191^{3} + 81\cdot 191^{4} + 105\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 6\cdot 191 + 81\cdot 191^{2} + 116\cdot 191^{3} + 52\cdot 191^{4} + 41\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 95 + 52\cdot 191 + 110\cdot 191^{2} + 165\cdot 191^{3} + 174\cdot 191^{4} + 48\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 120 + 45\cdot 191 + 167\cdot 191^{2} + 76\cdot 191^{3} + 178\cdot 191^{4} + 19\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 162 + 183\cdot 191 + 42\cdot 191^{2} + 146\cdot 191^{3} + 77\cdot 191^{4} + 70\cdot 191^{5} +O\left(191^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,3)(4,8)(6,7)$ |
| $(1,4,2,6)(3,8,5,7)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,6)(2,4)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,4)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ |
| $2$ | $8$ | $(1,5,6,8,2,3,4,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,4,5,2,7,6,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
