Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 98\cdot 103 + 44\cdot 103^{2} + 47\cdot 103^{3} + 36\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 45\cdot 103 + 43\cdot 103^{2} + 23\cdot 103^{3} + 47\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 87\cdot 103 + 21\cdot 103^{2} + 20\cdot 103^{3} + 20\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 31\cdot 103 + 91\cdot 103^{2} + 95\cdot 103^{3} + 13\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 + 17\cdot 103 + 20\cdot 103^{2} + 40\cdot 103^{3} + 99\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 + 56\cdot 103 + 30\cdot 103^{2} + 39\cdot 103^{3} + 76\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 84 + 19\cdot 103 + 39\cdot 103^{2} + 23\cdot 103^{3} + 79\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 93 + 55\cdot 103 + 17\cdot 103^{2} + 19\cdot 103^{3} + 39\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,6)(4,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,4,6)(2,3,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
