Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 80\cdot 107 + 20\cdot 107^{2} + 38\cdot 107^{3} + 8\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 30\cdot 107 + 24\cdot 107^{2} + 67\cdot 107^{3} + 45\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 36\cdot 107 + 33\cdot 107^{2} + 94\cdot 107^{3} + 23\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 + 101\cdot 107 + 91\cdot 107^{2} + 51\cdot 107^{3} + 45\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 + 5\cdot 107 + 15\cdot 107^{2} + 55\cdot 107^{3} + 61\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 79 + 70\cdot 107 + 73\cdot 107^{2} + 12\cdot 107^{3} + 83\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 89 + 76\cdot 107 + 82\cdot 107^{2} + 39\cdot 107^{3} + 61\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 100 + 26\cdot 107 + 86\cdot 107^{2} + 68\cdot 107^{3} + 98\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,4,2,8,3,5,7)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-1$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,6,4,2,8,3,5,7)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,2,5,6,8,7,4,3)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,3,4,7,8,6,5,2)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,7,5,3,8,2,4,6)$ | $-\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
