Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 36\cdot 113 + 81\cdot 113^{2} + 39\cdot 113^{3} + 105\cdot 113^{4} + 26\cdot 113^{5} + 22\cdot 113^{6} + 83\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 73\cdot 113 + 50\cdot 113^{2} + 100\cdot 113^{3} + 21\cdot 113^{4} + 76\cdot 113^{5} + 43\cdot 113^{6} + 12\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 63\cdot 113 + 110\cdot 113^{2} + 63\cdot 113^{3} + 68\cdot 113^{4} + 111\cdot 113^{5} + 112\cdot 113^{6} + 69\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 62\cdot 113 + 104\cdot 113^{2} + 10\cdot 113^{3} + 35\cdot 113^{4} + 15\cdot 113^{5} + 58\cdot 113^{6} + 68\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 + 50\cdot 113 + 8\cdot 113^{2} + 102\cdot 113^{3} + 77\cdot 113^{4} + 97\cdot 113^{5} + 54\cdot 113^{6} + 44\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 81 + 49\cdot 113 + 2\cdot 113^{2} + 49\cdot 113^{3} + 44\cdot 113^{4} + 113^{5} + 43\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 85 + 39\cdot 113 + 62\cdot 113^{2} + 12\cdot 113^{3} + 91\cdot 113^{4} + 36\cdot 113^{5} + 69\cdot 113^{6} + 100\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 108 + 76\cdot 113 + 31\cdot 113^{2} + 73\cdot 113^{3} + 7\cdot 113^{4} + 86\cdot 113^{5} + 90\cdot 113^{6} + 29\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,4,6,8,2,5,3)$ |
| $(3,6)(4,5)$ |
| $(1,5,8,4)(2,6,7,3)$ |
| $(2,7)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $4$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $4$ | $8$ | $(1,7,4,6,8,2,5,3)$ | $0$ |
| $4$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $0$ |
| $4$ | $8$ | $(1,7,4,3,8,2,5,6)$ | $0$ |
| $4$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
