Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 24\cdot 191 + 38\cdot 191^{2} + 179\cdot 191^{3} + 105\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 52 + 153\cdot 191 + 37\cdot 191^{2} + 145\cdot 191^{3} + 126\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 75 + 15\cdot 191 + 190\cdot 191^{2} + 50\cdot 191^{3} + 29\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 77 + 88\cdot 191 + 92\cdot 191^{2} + 87\cdot 191^{3} + 99\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 105 + 153\cdot 191 + 35\cdot 191^{2} + 82\cdot 191^{3} + 143\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 111 + 13\cdot 191 + 166\cdot 191^{2} + 82\cdot 191^{3} + 25\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 162 + 7\cdot 191 + 88\cdot 191^{2} + 123\cdot 191^{3} + 8\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 172 + 116\cdot 191 + 115\cdot 191^{2} + 12\cdot 191^{3} + 34\cdot 191^{4} +O\left(191^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,5,7,6,8,2,4)$ |
| $(1,5,6,2)(3,7,8,4)$ |
| $(1,6)(2,5)(3,8)(4,7)$ |
| $(1,2)(4,7)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,2)(4,7)(5,6)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,5,6,2)(3,7,8,4)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,3,6,8)(2,7,5,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,5,7,6,8,2,4)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,8,5,4,6,3,2,7)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
