Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 12.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 80\cdot 101 + 39\cdot 101^{2} + 77\cdot 101^{3} + 15\cdot 101^{4} + 20\cdot 101^{5} + 29\cdot 101^{6} + 39\cdot 101^{7} + 90\cdot 101^{8} + 3\cdot 101^{9} + 69\cdot 101^{10} + 98\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 101 + 37\cdot 101^{2} + 100\cdot 101^{3} + 93\cdot 101^{4} + 88\cdot 101^{5} + 77\cdot 101^{6} + 27\cdot 101^{7} + 93\cdot 101^{8} + 18\cdot 101^{9} + 55\cdot 101^{10} + 53\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 7\cdot 101 + 77\cdot 101^{2} + 57\cdot 101^{3} + 41\cdot 101^{4} + 82\cdot 101^{5} + 64\cdot 101^{7} + 45\cdot 101^{8} + 84\cdot 101^{9} + 81\cdot 101^{10} + 64\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 81\cdot 101 + 74\cdot 101^{2} + 24\cdot 101^{3} + 3\cdot 101^{4} + 74\cdot 101^{5} + 79\cdot 101^{6} + 64\cdot 101^{8} + 71\cdot 101^{9} + 60\cdot 101^{10} + 49\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 62 + 19\cdot 101 + 26\cdot 101^{2} + 76\cdot 101^{3} + 97\cdot 101^{4} + 26\cdot 101^{5} + 21\cdot 101^{6} + 100\cdot 101^{7} + 36\cdot 101^{8} + 29\cdot 101^{9} + 40\cdot 101^{10} + 51\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 + 93\cdot 101 + 23\cdot 101^{2} + 43\cdot 101^{3} + 59\cdot 101^{4} + 18\cdot 101^{5} + 100\cdot 101^{6} + 36\cdot 101^{7} + 55\cdot 101^{8} + 16\cdot 101^{9} + 19\cdot 101^{10} + 36\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 87 + 99\cdot 101 + 63\cdot 101^{2} + 7\cdot 101^{4} + 12\cdot 101^{5} + 23\cdot 101^{6} + 73\cdot 101^{7} + 7\cdot 101^{8} + 82\cdot 101^{9} + 45\cdot 101^{10} + 47\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 98 + 20\cdot 101 + 61\cdot 101^{2} + 23\cdot 101^{3} + 85\cdot 101^{4} + 80\cdot 101^{5} + 71\cdot 101^{6} + 61\cdot 101^{7} + 10\cdot 101^{8} + 97\cdot 101^{9} + 31\cdot 101^{10} + 2\cdot 101^{11} +O\left(101^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,6)(4,5)$ |
| $(2,7)(3,6)$ |
| $(1,5,6,2,8,4,3,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $4$ | $8$ | $(1,5,6,2,8,4,3,7)$ | $0$ |
| $4$ | $8$ | $(1,2,3,5,8,7,6,4)$ | $0$ |
| $4$ | $8$ | $(1,5,6,7,8,4,3,2)$ | $0$ |
| $4$ | $8$ | $(1,7,3,5,8,2,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
