Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 84\cdot 101 + 74\cdot 101^{2} + 58\cdot 101^{3} + 49\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 23\cdot 101 + 85\cdot 101^{2} + 45\cdot 101^{3} + 86\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 + 65\cdot 101 + 91\cdot 101^{2} + 44\cdot 101^{3} + 84\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 71 + 40\cdot 101 + 38\cdot 101^{2} + 21\cdot 101^{3} + 27\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 82 + 11\cdot 101 + 98\cdot 101^{2} + 76\cdot 101^{3} + 40\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 95 + 49\cdot 101 + 92\cdot 101^{2} + 44\cdot 101^{3} + 30\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 98 + 45\cdot 101 + 80\cdot 101^{2} + 90\cdot 101^{3} + 59\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 100 + 82\cdot 101 + 44\cdot 101^{2} + 20\cdot 101^{3} + 25\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,5,4)(2,7,8,6)$ |
| $(1,2)(3,6)(4,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,3,5,4)(2,7,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
