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 values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,3)(4,7)(5,6)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,7)(2,5)(3,6)(4,8)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(2,3)(5,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,5,8,6)(2,7,3,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,7,6,2,8,4,5,3)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,2,5,7,8,3,6,4)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
