Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 51\cdot 103 + 85\cdot 103^{2} + 74\cdot 103^{3} + 59\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 + 26\cdot 103 + 42\cdot 103^{2} + 34\cdot 103^{3} + 99\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 83\cdot 103 + 26\cdot 103^{2} + 63\cdot 103^{3} + 42\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 48 + 29\cdot 103 + 14\cdot 103^{2} + 89\cdot 103^{3} + 59\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 73\cdot 103 + 88\cdot 103^{2} + 13\cdot 103^{3} + 43\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 + 19\cdot 103 + 76\cdot 103^{2} + 39\cdot 103^{3} + 60\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 63 + 76\cdot 103 + 60\cdot 103^{2} + 68\cdot 103^{3} + 3\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 96 + 51\cdot 103 + 17\cdot 103^{2} + 28\cdot 103^{3} + 43\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,5,7,6,8,4,2,3)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-1$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,5,7,6,8,4,2,3)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
