Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 54\cdot 103 + 20\cdot 103^{2} + 23\cdot 103^{3} + 43\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 52\cdot 103 + 13\cdot 103^{2} + 16\cdot 103^{3} + 9\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 55\cdot 103 + 101\cdot 103^{2} + 51\cdot 103^{3} + 26\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 + 74\cdot 103 + 68\cdot 103^{2} + 99\cdot 103^{3} + 52\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 24\cdot 103 + 22\cdot 103^{2} + 38\cdot 103^{3} + 14\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 25\cdot 103 + 67\cdot 103^{3} + 100\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 74 + 81\cdot 103 + 11\cdot 103^{2} + 103^{3} + 38\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 75 + 44\cdot 103 + 70\cdot 103^{2} + 11\cdot 103^{3} + 24\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,7)(5,6)$ |
| $(1,2,4,6)(3,5,7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,4,6)(3,5,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
