Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 74\cdot 113 + 84\cdot 113^{2} + 22\cdot 113^{3} + 112\cdot 113^{4} + 13\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 + 110\cdot 113 + 96\cdot 113^{2} + 65\cdot 113^{3} + 69\cdot 113^{4} + 110\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 58 + 92\cdot 113 + 113^{2} + 60\cdot 113^{3} + 62\cdot 113^{4} + 93\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 62 + 56\cdot 113 + 48\cdot 113^{2} + 98\cdot 113^{3} + 87\cdot 113^{4} + 103\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 63 + 19\cdot 113 + 49\cdot 113^{2} + 41\cdot 113^{3} + 36\cdot 113^{4} + 94\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 74 + 102\cdot 113 + 76\cdot 113^{2} + 46\cdot 113^{3} + 46\cdot 113^{4} + 30\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 75 + 60\cdot 113 + 64\cdot 113^{2} + 87\cdot 113^{3} + 99\cdot 113^{4} + 6\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 78 + 48\cdot 113 + 29\cdot 113^{2} + 29\cdot 113^{3} + 50\cdot 113^{4} + 111\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,3,4,6,2,7,8,5)$ |
| $(1,4)(2,8)(5,6)$ |
| $(1,8,2,4)(3,5,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,8)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,2,8)(3,6,7,5)$ | $0$ |
| $2$ | $8$ | $(1,3,4,6,2,7,8,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,6,8,3,2,5,4,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
