Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 12.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 + 114\cdot 181 + 136\cdot 181^{2} + 53\cdot 181^{3} + 45\cdot 181^{4} + 16\cdot 181^{5} + 135\cdot 181^{6} + 50\cdot 181^{7} + 5\cdot 181^{8} + 97\cdot 181^{9} + 48\cdot 181^{10} + 32\cdot 181^{11} +O\left(181^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 + 53\cdot 181 + 52\cdot 181^{2} + 132\cdot 181^{3} + 16\cdot 181^{4} + 24\cdot 181^{5} + 113\cdot 181^{6} + 108\cdot 181^{7} + 134\cdot 181^{8} + 135\cdot 181^{9} + 108\cdot 181^{10} + 164\cdot 181^{11} +O\left(181^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 + 23\cdot 181 + 109\cdot 181^{2} + 160\cdot 181^{3} + 99\cdot 181^{4} + 126\cdot 181^{5} + 39\cdot 181^{6} + 135\cdot 181^{7} + 79\cdot 181^{8} + 33\cdot 181^{9} + 91\cdot 181^{10} + 138\cdot 181^{11} +O\left(181^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 80 + 178\cdot 181 + 135\cdot 181^{2} + 109\cdot 181^{3} + 138\cdot 181^{4} + 74\cdot 181^{5} + 175\cdot 181^{6} + 154\cdot 181^{7} + 112\cdot 181^{8} + 21\cdot 181^{9} + 66\cdot 181^{10} + 47\cdot 181^{11} +O\left(181^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 101 + 2\cdot 181 + 45\cdot 181^{2} + 71\cdot 181^{3} + 42\cdot 181^{4} + 106\cdot 181^{5} + 5\cdot 181^{6} + 26\cdot 181^{7} + 68\cdot 181^{8} + 159\cdot 181^{9} + 114\cdot 181^{10} + 133\cdot 181^{11} +O\left(181^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 109 + 157\cdot 181 + 71\cdot 181^{2} + 20\cdot 181^{3} + 81\cdot 181^{4} + 54\cdot 181^{5} + 141\cdot 181^{6} + 45\cdot 181^{7} + 101\cdot 181^{8} + 147\cdot 181^{9} + 89\cdot 181^{10} + 42\cdot 181^{11} +O\left(181^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 126 + 127\cdot 181 + 128\cdot 181^{2} + 48\cdot 181^{3} + 164\cdot 181^{4} + 156\cdot 181^{5} + 67\cdot 181^{6} + 72\cdot 181^{7} + 46\cdot 181^{8} + 45\cdot 181^{9} + 72\cdot 181^{10} + 16\cdot 181^{11} +O\left(181^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 140 + 66\cdot 181 + 44\cdot 181^{2} + 127\cdot 181^{3} + 135\cdot 181^{4} + 164\cdot 181^{5} + 45\cdot 181^{6} + 130\cdot 181^{7} + 175\cdot 181^{8} + 83\cdot 181^{9} + 132\cdot 181^{10} + 148\cdot 181^{11} +O\left(181^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,6)(4,5)$ |
| $(2,7)(3,6)$ |
| $(1,3,7,5,8,6,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $4$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $0$ |
| $4$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $0$ |
| $4$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $0$ |
| $4$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
