Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 12.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 + 2\cdot 103 + 11\cdot 103^{2} + 33\cdot 103^{3} + 76\cdot 103^{4} + 100\cdot 103^{5} + 82\cdot 103^{6} + 4\cdot 103^{7} + 97\cdot 103^{8} + 98\cdot 103^{9} + 78\cdot 103^{10} + 77\cdot 103^{11} +O\left(103^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 46\cdot 103 + 99\cdot 103^{2} + 24\cdot 103^{3} + 75\cdot 103^{4} + 29\cdot 103^{5} + 23\cdot 103^{6} + 83\cdot 103^{7} + 32\cdot 103^{8} + 79\cdot 103^{9} + 40\cdot 103^{10} + 22\cdot 103^{11} +O\left(103^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 11\cdot 103 + 99\cdot 103^{2} + 5\cdot 103^{3} + 7\cdot 103^{4} + 34\cdot 103^{5} + 34\cdot 103^{6} + 37\cdot 103^{7} + 41\cdot 103^{8} + 101\cdot 103^{9} + 7\cdot 103^{10} + 92\cdot 103^{11} +O\left(103^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 93\cdot 103 + 22\cdot 103^{2} + 101\cdot 103^{3} + 52\cdot 103^{4} + 15\cdot 103^{5} + 82\cdot 103^{6} + 5\cdot 103^{7} + 18\cdot 103^{8} + 103^{9} + 2\cdot 103^{10} + 36\cdot 103^{11} +O\left(103^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 9\cdot 103 + 80\cdot 103^{2} + 103^{3} + 50\cdot 103^{4} + 87\cdot 103^{5} + 20\cdot 103^{6} + 97\cdot 103^{7} + 84\cdot 103^{8} + 101\cdot 103^{9} + 100\cdot 103^{10} + 66\cdot 103^{11} +O\left(103^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 73 + 91\cdot 103 + 3\cdot 103^{2} + 97\cdot 103^{3} + 95\cdot 103^{4} + 68\cdot 103^{5} + 68\cdot 103^{6} + 65\cdot 103^{7} + 61\cdot 103^{8} + 103^{9} + 95\cdot 103^{10} + 10\cdot 103^{11} +O\left(103^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 76 + 56\cdot 103 + 3\cdot 103^{2} + 78\cdot 103^{3} + 27\cdot 103^{4} + 73\cdot 103^{5} + 79\cdot 103^{6} + 19\cdot 103^{7} + 70\cdot 103^{8} + 23\cdot 103^{9} + 62\cdot 103^{10} + 80\cdot 103^{11} +O\left(103^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 83 + 100\cdot 103 + 91\cdot 103^{2} + 69\cdot 103^{3} + 26\cdot 103^{4} + 2\cdot 103^{5} + 20\cdot 103^{6} + 98\cdot 103^{7} + 5\cdot 103^{8} + 4\cdot 103^{9} + 24\cdot 103^{10} + 25\cdot 103^{11} +O\left(103^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,2,4,8,6,7,5)$ |
| $(1,5,8,4)(2,6,7,3)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,7)(2,8)(4,5)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,2,4,8,6,7,5)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,6,2,5,8,3,7,4)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
