Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 46\cdot 103 + 47\cdot 103^{2} + 91\cdot 103^{3} + 6\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 52\cdot 103 + 21\cdot 103^{2} + 29\cdot 103^{3} + 99\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 + 30\cdot 103 + 64\cdot 103^{2} + 85\cdot 103^{3} + 56\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 41\cdot 103 + 41\cdot 103^{2} + 52\cdot 103^{3} + 66\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 + 32\cdot 103 + 14\cdot 103^{2} + 65\cdot 103^{3} + 29\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 73 + 9\cdot 103 + 29\cdot 103^{2} + 91\cdot 103^{3} + 55\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 83 + 8\cdot 103 + 77\cdot 103^{2} + 23\cdot 103^{3} + 12\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 97 + 87\cdot 103 + 13\cdot 103^{2} + 76\cdot 103^{3} + 84\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,5)(3,6)(4,8)$ |
| $(1,6)(2,3)(5,8)$ |
| $(1,5,8,6)(2,7,3,4)$ |
| $(1,8)(2,3)(4,7)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,3)(4,7)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,3)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,6)(2,7,3,4)$ | $0$ |
| $2$ | $8$ | $(1,7,6,2,8,4,5,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,2,5,7,8,3,6,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
