Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 33\cdot 107 + 90\cdot 107^{2} + 79\cdot 107^{3} + 91\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 49\cdot 107 + 61\cdot 107^{2} + 56\cdot 107^{3} + 65\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 62\cdot 107 + 99\cdot 107^{2} + 103\cdot 107^{3} + 99\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 78\cdot 107 + 70\cdot 107^{2} + 80\cdot 107^{3} + 73\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 66 + 28\cdot 107 + 36\cdot 107^{2} + 26\cdot 107^{3} + 33\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 76 + 44\cdot 107 + 7\cdot 107^{2} + 3\cdot 107^{3} + 7\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 86 + 57\cdot 107 + 45\cdot 107^{2} + 50\cdot 107^{3} + 41\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 96 + 73\cdot 107 + 16\cdot 107^{2} + 27\cdot 107^{3} + 15\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,3,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
