Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 42\cdot 103 + 20\cdot 103^{2} + 95\cdot 103^{3} + 60\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 64\cdot 103 + 79\cdot 103^{2} + 33\cdot 103^{3} + 40\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 2\cdot 103 + 101\cdot 103^{2} + 77\cdot 103^{3} + 7\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 + 87\cdot 103 + 96\cdot 103^{2} + 3\cdot 103^{3} + 44\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 82\cdot 103 + 36\cdot 103^{2} + 16\cdot 103^{3} + 36\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 + 66\cdot 103 + 10\cdot 103^{2} + 103^{3} + 78\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 88 + 31\cdot 103 + 14\cdot 103^{2} + 40\cdot 103^{3} + 31\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 90 + 35\cdot 103 + 52\cdot 103^{2} + 40\cdot 103^{3} + 10\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,4,2)(3,7,8,5)$ |
| $(1,8,4,3)(2,5,6,7)$ |
| $(1,4)(2,6)(3,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $-2$ |
| $2$ | $4$ | $(1,8,4,3)(2,5,6,7)$ | $0$ |
| $2$ | $4$ | $(1,6,4,2)(3,7,8,5)$ | $0$ |
| $2$ | $4$ | $(1,5,4,7)(2,3,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
