Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 12.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 95\cdot 101 + 39\cdot 101^{2} + 16\cdot 101^{3} + 18\cdot 101^{4} + 16\cdot 101^{5} + 51\cdot 101^{6} + 73\cdot 101^{7} + 6\cdot 101^{8} + 63\cdot 101^{9} + 8\cdot 101^{10} + 7\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 80\cdot 101 + 29\cdot 101^{2} + 24\cdot 101^{3} + 48\cdot 101^{4} + 92\cdot 101^{5} + 4\cdot 101^{6} + 39\cdot 101^{7} + 47\cdot 101^{8} + 101^{9} + 49\cdot 101^{10} + 65\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 97\cdot 101 + 21\cdot 101^{2} + 46\cdot 101^{3} + 58\cdot 101^{4} + 9\cdot 101^{5} + 47\cdot 101^{6} + 7\cdot 101^{7} + 8\cdot 101^{8} + 50\cdot 101^{9} + 51\cdot 101^{10} + 31\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 65\cdot 101 + 83\cdot 101^{2} + 95\cdot 101^{3} + 83\cdot 101^{4} + 42\cdot 101^{5} + 64\cdot 101^{6} + 54\cdot 101^{7} + 37\cdot 101^{8} + 2\cdot 101^{9} + 71\cdot 101^{10} + 38\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 67 + 35\cdot 101 + 17\cdot 101^{2} + 5\cdot 101^{3} + 17\cdot 101^{4} + 58\cdot 101^{5} + 36\cdot 101^{6} + 46\cdot 101^{7} + 63\cdot 101^{8} + 98\cdot 101^{9} + 29\cdot 101^{10} + 62\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 81 + 3\cdot 101 + 79\cdot 101^{2} + 54\cdot 101^{3} + 42\cdot 101^{4} + 91\cdot 101^{5} + 53\cdot 101^{6} + 93\cdot 101^{7} + 92\cdot 101^{8} + 50\cdot 101^{9} + 49\cdot 101^{10} + 69\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 83 + 20\cdot 101 + 71\cdot 101^{2} + 76\cdot 101^{3} + 52\cdot 101^{4} + 8\cdot 101^{5} + 96\cdot 101^{6} + 61\cdot 101^{7} + 53\cdot 101^{8} + 99\cdot 101^{9} + 51\cdot 101^{10} + 35\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 85 + 5\cdot 101 + 61\cdot 101^{2} + 84\cdot 101^{3} + 82\cdot 101^{4} + 84\cdot 101^{5} + 49\cdot 101^{6} + 27\cdot 101^{7} + 94\cdot 101^{8} + 37\cdot 101^{9} + 92\cdot 101^{10} + 93\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,2,4,8,3,7,5)$ |
| $(3,6)(4,5)$ |
| $(2,7)(4,5)$ |
| $(1,8)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $4$ | $8$ | $(1,6,2,4,8,3,7,5)$ | $0$ |
| $4$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $0$ |
| $4$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $0$ |
| $4$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
