Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 98\cdot 107 + 28\cdot 107^{2} + 9\cdot 107^{3} + 67\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 70\cdot 107 + 72\cdot 107^{2} + 34\cdot 107^{3} + 71\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 + 57\cdot 107 + 13\cdot 107^{2} + 60\cdot 107^{3} + 58\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 + 102\cdot 107 + 99\cdot 107^{2} + 59\cdot 107^{3} + 41\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 55\cdot 107 + 12\cdot 107^{2} + 57\cdot 107^{3} + 96\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 68 + 31\cdot 107 + 101\cdot 107^{2} + 35\cdot 107^{3} + 39\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 73 + 60\cdot 107 + 32\cdot 107^{2} + 68\cdot 107^{3} + 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 99 + 58\cdot 107 + 66\cdot 107^{2} + 102\cdot 107^{3} + 51\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,2,6,7,4,3,8)$ |
| $(1,7)(2,3)(4,5)(6,8)$ |
| $(1,3,7,2)(4,6,5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $-1$ |
| $1$ | $4$ | $(1,2,7,3)(4,8,5,6)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,3,7,2)(4,6,5,8)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,5,2,6,7,4,3,8)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,6,3,5,7,8,2,4)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,4,2,8,7,5,3,6)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,8,3,4,7,6,2,5)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
