Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 44\cdot 103 + 93\cdot 103^{2} + 93\cdot 103^{3} + 71\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 96\cdot 103 + 93\cdot 103^{2} + 49\cdot 103^{3} + 30\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 15\cdot 103 + 6\cdot 103^{2} + 82\cdot 103^{3} + 88\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 52\cdot 103 + 90\cdot 103^{2} + 19\cdot 103^{3} + 88\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 + 50\cdot 103 + 12\cdot 103^{2} + 83\cdot 103^{3} + 14\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 87 + 87\cdot 103 + 96\cdot 103^{2} + 20\cdot 103^{3} + 14\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 91 + 6\cdot 103 + 9\cdot 103^{2} + 53\cdot 103^{3} + 72\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 92 + 58\cdot 103 + 9\cdot 103^{2} + 9\cdot 103^{3} + 31\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,5)(4,8,7,6)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,7)(3,8)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,5)(4,8,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
