Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 68\cdot 101 + 91\cdot 101^{2} + 55\cdot 101^{3} + 85\cdot 101^{4} + 98\cdot 101^{5} + 91\cdot 101^{6} + 86\cdot 101^{7} + 43\cdot 101^{8} + 59\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 93\cdot 101 + 17\cdot 101^{2} + 54\cdot 101^{3} + 9\cdot 101^{4} + 10\cdot 101^{5} + 9\cdot 101^{6} + 91\cdot 101^{7} + 33\cdot 101^{8} + 55\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 + 44\cdot 101 + 7\cdot 101^{2} + 33\cdot 101^{3} + 33\cdot 101^{4} + 42\cdot 101^{5} + 15\cdot 101^{6} + 66\cdot 101^{7} + 52\cdot 101^{8} + 4\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 71\cdot 101 + 40\cdot 101^{2} + 90\cdot 101^{3} + 82\cdot 101^{4} + 8\cdot 101^{5} + 39\cdot 101^{6} + 60\cdot 101^{7} + 98\cdot 101^{8} + 87\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 29\cdot 101 + 60\cdot 101^{2} + 10\cdot 101^{3} + 18\cdot 101^{4} + 92\cdot 101^{5} + 61\cdot 101^{6} + 40\cdot 101^{7} + 2\cdot 101^{8} + 13\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 63 + 56\cdot 101 + 93\cdot 101^{2} + 67\cdot 101^{3} + 67\cdot 101^{4} + 58\cdot 101^{5} + 85\cdot 101^{6} + 34\cdot 101^{7} + 48\cdot 101^{8} + 96\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 7\cdot 101 + 83\cdot 101^{2} + 46\cdot 101^{3} + 91\cdot 101^{4} + 90\cdot 101^{5} + 91\cdot 101^{6} + 9\cdot 101^{7} + 67\cdot 101^{8} + 45\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 95 + 32\cdot 101 + 9\cdot 101^{2} + 45\cdot 101^{3} + 15\cdot 101^{4} + 2\cdot 101^{5} + 9\cdot 101^{6} + 14\cdot 101^{7} + 57\cdot 101^{8} + 41\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(3,6)(4,5)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(2,7)(3,6)$ |
| $(1,3,7,5,8,6,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $4$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $0$ |
| $4$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $0$ |
| $4$ | $8$ | $(1,6,7,5,8,3,2,4)$ | $0$ |
| $4$ | $8$ | $(1,5,2,6,8,4,7,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
