Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 17\cdot 107 + 79\cdot 107^{2} + 96\cdot 107^{3} + 31\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 8\cdot 107 + 80\cdot 107^{2} + 15\cdot 107^{3} + 52\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 81\cdot 107 + 27\cdot 107^{2} + 104\cdot 107^{3} + 21\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 85\cdot 107 + 54\cdot 107^{2} + 8\cdot 107^{3} + 44\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 81\cdot 107 + 45\cdot 107^{2} + 16\cdot 107^{3} + 16\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 75 + 7\cdot 107 + 55\cdot 107^{2} + 2\cdot 107^{3} + 28\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 98 + 72\cdot 107 + 85\cdot 107^{2} + 84\cdot 107^{3} + 24\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 105 + 73\cdot 107 + 106\cdot 107^{2} + 98\cdot 107^{3} + 101\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,6,2)(3,7,4,5)$ |
| $(1,6)(2,8)(3,4)(5,7)$ |
| $(2,8)(3,5)(4,7)$ |
| $(1,4,6,3)(2,5,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,8)(3,4)(5,7)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(2,8)(3,5)(4,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,8,6,2)(3,7,4,5)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,4,6,3)(2,5,8,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,7,8,4,6,5,2,3)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,5,8,3,6,7,2,4)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
