Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 8\cdot 103 + 22\cdot 103^{2} + 41\cdot 103^{3} + 58\cdot 103^{4} + 86\cdot 103^{5} + 89\cdot 103^{6} + 54\cdot 103^{7} + 65\cdot 103^{8} + 19\cdot 103^{9} + 14\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 + 102\cdot 103 + 60\cdot 103^{2} + 71\cdot 103^{3} + 65\cdot 103^{4} + 86\cdot 103^{5} + 81\cdot 103^{6} + 52\cdot 103^{7} + 83\cdot 103^{8} + 70\cdot 103^{9} + 53\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 35\cdot 103 + 78\cdot 103^{2} + 95\cdot 103^{3} + 62\cdot 103^{4} + 6\cdot 103^{5} + 66\cdot 103^{6} + 17\cdot 103^{7} + 57\cdot 103^{8} + 64\cdot 103^{9} + 51\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 61\cdot 103 + 16\cdot 103^{2} + 9\cdot 103^{3} + 39\cdot 103^{4} + 100\cdot 103^{5} + 41\cdot 103^{6} + 21\cdot 103^{7} + 63\cdot 103^{8} + 27\cdot 103^{9} + 76\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 84 + 97\cdot 103 + 67\cdot 103^{2} + 53\cdot 103^{3} + 48\cdot 103^{4} + 94\cdot 103^{5} + 59\cdot 103^{6} + 63\cdot 103^{7} + 38\cdot 103^{8} + 58\cdot 103^{9} + 91\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 86 + 7\cdot 103 + 76\cdot 103^{2} + 97\cdot 103^{3} + 89\cdot 103^{4} + 70\cdot 103^{5} + 96\cdot 103^{6} + 25\cdot 103^{7} + 76\cdot 103^{8} + 78\cdot 103^{9} + 33\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 89 + 30\cdot 103 + 99\cdot 103^{2} + 102\cdot 103^{3} + 84\cdot 103^{4} + 4\cdot 103^{5} + 40\cdot 103^{6} + 30\cdot 103^{7} + 102\cdot 103^{8} + 63\cdot 103^{9} + 5\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 99 + 67\cdot 103 + 93\cdot 103^{2} + 42\cdot 103^{3} + 65\cdot 103^{4} + 64\cdot 103^{5} + 38\cdot 103^{6} + 42\cdot 103^{7} + 28\cdot 103^{8} + 28\cdot 103^{9} + 85\cdot 103^{10} +O\left(103^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(4,5)$ |
| $(1,4)(2,3)(5,7)(6,8)$ |
| $(2,6)(4,5)$ |
| $(1,3)(7,8)$ |
| $(1,7)(3,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-4$ |
| $2$ | $2$ | $(2,6)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(2,4)(5,6)$ | $2$ |
| $4$ | $2$ | $(1,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $-2$ |
| $4$ | $4$ | $(1,4,7,5)(2,8,6,3)$ | $0$ |
| $4$ | $4$ | $(1,5,7,4)(2,8,6,3)$ | $0$ |
| $4$ | $4$ | $(1,3,7,8)(2,5,6,4)$ | $0$ |
| $8$ | $4$ | $(1,4,3,2)(5,8,6,7)$ | $0$ |
| $8$ | $4$ | $(1,5,8,6)(2,7,4,3)$ | $0$ |
| $8$ | $4$ | $(1,7)(2,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
