Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 45\cdot 101 + 69\cdot 101^{2} + 23\cdot 101^{3} + 18\cdot 101^{4} + 83\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 22\cdot 101 + 73\cdot 101^{2} + 38\cdot 101^{3} + 90\cdot 101^{4} + 13\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 17\cdot 101 + 3\cdot 101^{2} + 90\cdot 101^{3} + 81\cdot 101^{4} + 55\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 31\cdot 101 + 75\cdot 101^{2} + 11\cdot 101^{3} + 46\cdot 101^{4} + 19\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 69\cdot 101 + 25\cdot 101^{2} + 89\cdot 101^{3} + 54\cdot 101^{4} + 81\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 73 + 83\cdot 101 + 97\cdot 101^{2} + 10\cdot 101^{3} + 19\cdot 101^{4} + 45\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 79 + 78\cdot 101 + 27\cdot 101^{2} + 62\cdot 101^{3} + 10\cdot 101^{4} + 87\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 85 + 55\cdot 101 + 31\cdot 101^{2} + 77\cdot 101^{3} + 82\cdot 101^{4} + 17\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
