Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 39 + 15\cdot 103 + 96\cdot 103^{2} + 79\cdot 103^{3} + 38\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 + 74\cdot 103 + 72\cdot 103^{2} + 90\cdot 103^{3} + 41\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 73 + 47\cdot 103 + 20\cdot 103^{2} + 14\cdot 103^{3} + 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 81 + 102\cdot 103 + 32\cdot 103^{2} + 28\cdot 103^{3} + 71\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 90 + 83\cdot 103 + 65\cdot 103^{2} + 54\cdot 103^{3} + 5\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 93 + 19\cdot 103 + 19\cdot 103^{2} + 47\cdot 103^{3} + 74\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 96 + 68\cdot 103 + 72\cdot 103^{2} + 7\cdot 103^{3} + 72\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 98 + 101\cdot 103 + 31\cdot 103^{2} + 89\cdot 103^{3} + 3\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,6,7,3,8,5,2)$ |
| $(1,3)(2,7)(4,8)(5,6)$ |
| $(1,5,3,6)(2,8,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $-1$ |
| $1$ | $4$ | $(1,6,3,5)(2,4,7,8)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,5,3,6)(2,8,7,4)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,4,6,7,3,8,5,2)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,7,5,4,3,2,6,8)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,8,6,2,3,4,5,7)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,2,5,8,3,7,6,4)$ | $-\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
