Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 + 103\cdot 113 + 63\cdot 113^{2} + 96\cdot 113^{3} + 113^{4} + 22\cdot 113^{5} + 99\cdot 113^{6} + 10\cdot 113^{7} + 43\cdot 113^{8} +O\left(113^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 83\cdot 113^{2} + 109\cdot 113^{3} + 2\cdot 113^{4} + 83\cdot 113^{5} + 9\cdot 113^{6} + 16\cdot 113^{7} + 17\cdot 113^{8} +O\left(113^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 30\cdot 113 + 104\cdot 113^{2} + 101\cdot 113^{3} + 101\cdot 113^{4} + 51\cdot 113^{5} + 61\cdot 113^{6} + 84\cdot 113^{7} + 21\cdot 113^{8} +O\left(113^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 26\cdot 113 + 75\cdot 113^{2} + 10\cdot 113^{3} + 90\cdot 113^{4} + 111\cdot 113^{5} + 43\cdot 113^{6} + 55\cdot 113^{7} + 101\cdot 113^{8} +O\left(113^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 67 + 86\cdot 113 + 37\cdot 113^{2} + 102\cdot 113^{3} + 22\cdot 113^{4} + 113^{5} + 69\cdot 113^{6} + 57\cdot 113^{7} + 11\cdot 113^{8} +O\left(113^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 74 + 82\cdot 113 + 8\cdot 113^{2} + 11\cdot 113^{3} + 11\cdot 113^{4} + 61\cdot 113^{5} + 51\cdot 113^{6} + 28\cdot 113^{7} + 91\cdot 113^{8} +O\left(113^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 81 + 112\cdot 113 + 29\cdot 113^{2} + 3\cdot 113^{3} + 110\cdot 113^{4} + 29\cdot 113^{5} + 103\cdot 113^{6} + 96\cdot 113^{7} + 95\cdot 113^{8} +O\left(113^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 83 + 9\cdot 113 + 49\cdot 113^{2} + 16\cdot 113^{3} + 111\cdot 113^{4} + 90\cdot 113^{5} + 13\cdot 113^{6} + 102\cdot 113^{7} + 69\cdot 113^{8} +O\left(113^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,8)(4,5)$ |
| $(1,7,4,3,8,2,5,6)$ |
| $(2,7)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,7,4,3,8,2,5,6)$ | $0$ |
| $2$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $0$ |
| $2$ | $8$ | $(1,2,5,3,8,7,4,6)$ | $0$ |
| $2$ | $8$ | $(1,3,4,2,8,6,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
