Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 61\cdot 103 + 68\cdot 103^{2} + 85\cdot 103^{3} + 80\cdot 103^{4} + 29\cdot 103^{5} + 30\cdot 103^{6} + 23\cdot 103^{7} + 32\cdot 103^{8} + 11\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 28\cdot 103 + 98\cdot 103^{2} + 62\cdot 103^{3} + 96\cdot 103^{4} + 52\cdot 103^{5} + 69\cdot 103^{6} + 47\cdot 103^{7} + 103^{8} + 66\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 75\cdot 103 + 73\cdot 103^{2} + 75\cdot 103^{3} + 31\cdot 103^{4} + 18\cdot 103^{5} + 35\cdot 103^{6} + 12\cdot 103^{7} + 84\cdot 103^{8} + 26\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 53 + 47\cdot 103 + 5\cdot 103^{2} + 10\cdot 103^{3} + 68\cdot 103^{4} + 102\cdot 103^{5} + 48\cdot 103^{6} + 53\cdot 103^{7} + 52\cdot 103^{8} + 45\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 + 44\cdot 103 + 75\cdot 103^{2} + 17\cdot 103^{3} + 46\cdot 103^{4} + 60\cdot 103^{5} + 3\cdot 103^{6} + 44\cdot 103^{7} + 46\cdot 103^{8} + 101\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 77 + 11\cdot 103 + 9\cdot 103^{2} + 45\cdot 103^{3} + 41\cdot 103^{4} + 31\cdot 103^{5} + 98\cdot 103^{6} + 40\cdot 103^{7} + 55\cdot 103^{8} + 87\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 80 + 62\cdot 103 + 28\cdot 103^{2} + 56\cdot 103^{3} + 87\cdot 103^{4} + 54\cdot 103^{5} + 26\cdot 103^{6} + 25\cdot 103^{7} + 4\cdot 103^{8} + 103^{9} +O\left(103^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 84 + 80\cdot 103 + 52\cdot 103^{2} + 58\cdot 103^{3} + 62\cdot 103^{4} + 61\cdot 103^{5} + 99\cdot 103^{6} + 61\cdot 103^{7} + 32\cdot 103^{8} + 72\cdot 103^{9} +O\left(103^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)$ |
| $(2,3)(4,7)$ |
| $(1,5)(2,4)(3,7)(6,8)$ |
| $(4,7)(5,6)$ |
| $(1,8)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,3)(4,7)(5,6)$ |
$-4$ |
| $2$ |
$2$ |
$(1,2)(3,8)(4,5)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,3)$ |
$0$ |
| $2$ |
$2$ |
$(1,2)(3,8)(4,6)(5,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,8)$ |
$2$ |
| $4$ |
$2$ |
$(2,3)(4,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,4)(3,7)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,7)(3,4)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,8)(4,7)(5,6)$ |
$-2$ |
| $4$ |
$4$ |
$(1,6,8,5)(2,4,3,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,8,5)(2,7,3,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,8,3)(4,5,7,6)$ |
$0$ |
| $8$ |
$4$ |
$(1,4,2,5)(3,6,8,7)$ |
$0$ |
| $8$ |
$4$ |
$(1,2,8,3)(4,7)$ |
$0$ |
| $8$ |
$4$ |
$(1,7,2,5)(3,6,8,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
