Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 100\cdot 103 + 99\cdot 103^{2} + 16\cdot 103^{3} + 74\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 57\cdot 103 + 69\cdot 103^{2} + 52\cdot 103^{3} + 28\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 85\cdot 103 + 29\cdot 103^{2} + 91\cdot 103^{3} + 57\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 17\cdot 103 + 43\cdot 103^{2} + 101\cdot 103^{3} + 74\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 60 + 85\cdot 103 + 59\cdot 103^{2} + 103^{3} + 28\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 72 + 17\cdot 103 + 73\cdot 103^{2} + 11\cdot 103^{3} + 45\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 82 + 45\cdot 103 + 33\cdot 103^{2} + 50\cdot 103^{3} + 74\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 87 + 2\cdot 103 + 3\cdot 103^{2} + 86\cdot 103^{3} + 28\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
