Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 + 93\cdot 197 + 77\cdot 197^{2} + 102\cdot 197^{3} + 196\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 39 + 150\cdot 197 + 15\cdot 197^{2} + 79\cdot 197^{3} + 172\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 + 31\cdot 197 + 175\cdot 197^{2} + 38\cdot 197^{3} + 183\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 101 + 8\cdot 197 + 162\cdot 197^{2} + 139\cdot 197^{3} + 29\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 104 + 28\cdot 197 + 20\cdot 197^{2} + 173\cdot 197^{3} + 103\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 112 + 79\cdot 197 + 82\cdot 197^{2} + 181\cdot 197^{3} + 119\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 161 + 15\cdot 197 + 72\cdot 197^{2} + 167\cdot 197^{3} + 47\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 191 + 183\cdot 197 + 182\cdot 197^{2} + 102\cdot 197^{3} + 131\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(3,8)(4,7)$ |
| $(1,8)(2,7)(3,4)(5,6)$ |
| $(1,4)(2,5)(3,8)(6,7)$ |
| $(1,6,4,7)(2,3,5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,6)(3,8)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,6,4,7)(2,3,5,8)$ | $0$ |
| $2$ | $8$ | $(1,3,7,2,4,8,6,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,2,6,3,4,5,7,8)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
