Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 87\cdot 107 + 23\cdot 107^{2} + 3\cdot 107^{3} + 37\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 64\cdot 107 + 11\cdot 107^{2} + 30\cdot 107^{3} + 36\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 72\cdot 107 + 29\cdot 107^{2} + 23\cdot 107^{3} + 26\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 61 + 71\cdot 107 + 7\cdot 107^{2} + 14\cdot 107^{3} + 94\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 + 67\cdot 107 + 43\cdot 107^{2} + 81\cdot 107^{3} + 97\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 84 + 4\cdot 107 + 61\cdot 107^{2} + 21\cdot 107^{3} + 47\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 88 + 100\cdot 107 + 48\cdot 107^{2} + 59\cdot 107^{3} + 78\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 89 + 66\cdot 107 + 94\cdot 107^{2} + 87\cdot 107^{3} + 10\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,7,6,8)(2,4,5,3)$ |
| $(1,7)(2,5)(6,8)$ |
| $(1,6)(2,5)(3,4)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,6,8)(2,4,5,3)$ | $0$ |
| $2$ | $8$ | $(1,5,8,4,6,2,7,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,4,7,5,6,3,8,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
