Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 55\cdot 103 + 55\cdot 103^{2} + 97\cdot 103^{3} + 36\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 53\cdot 103 + 92\cdot 103^{2} + 23\cdot 103^{3} + 74\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 42 + 88\cdot 103 + 89\cdot 103^{2} + 77\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 25\cdot 103 + 28\cdot 103^{2} + 4\cdot 103^{3} + 75\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 54 + 50\cdot 103 + 95\cdot 103^{2} + 77\cdot 103^{3} + 5\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 69 + 37\cdot 103 + 88\cdot 103^{2} + 89\cdot 103^{3} + 67\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 83 + 2\cdot 103 + 79\cdot 103^{2} + 84\cdot 103^{3} + 3\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 96 + 98\cdot 103 + 88\cdot 103^{2} + 32\cdot 103^{3} + 71\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,3,7)(2,4,5,6)$ |
| $(1,2,8,4,3,5,7,6)$ |
| $(1,3)(2,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $-1$ |
| $1$ | $4$ | $(1,8,3,7)(2,4,5,6)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,7,3,8)(2,6,5,4)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,2,8,4,3,5,7,6)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,7,2,3,6,8,5)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,5,8,6,3,2,7,4)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,6,7,5,3,4,8,2)$ | $-\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
