Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 281 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 + 27\cdot 281 + 19\cdot 281^{2} + 18\cdot 281^{3} + 184\cdot 281^{4} + 173\cdot 281^{5} + 71\cdot 281^{6} + 156\cdot 281^{7} + 23\cdot 281^{8} +O\left(281^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 112 + 175\cdot 281 + 194\cdot 281^{2} + 155\cdot 281^{3} + 45\cdot 281^{4} + 269\cdot 281^{5} + 90\cdot 281^{6} + 25\cdot 281^{7} + 225\cdot 281^{8} +O\left(281^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 135 + 224\cdot 281 + 24\cdot 281^{2} + 46\cdot 281^{3} + 23\cdot 281^{4} + 88\cdot 281^{5} + 109\cdot 281^{6} + 253\cdot 281^{7} + 14\cdot 281^{8} +O\left(281^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 164 + 73\cdot 281 + 175\cdot 281^{2} + 219\cdot 281^{3} + 152\cdot 281^{4} + 89\cdot 281^{5} + 226\cdot 281^{6} + 151\cdot 281^{7} + 163\cdot 281^{8} +O\left(281^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 173 + 36\cdot 281 + 161\cdot 281^{2} + 178\cdot 281^{3} + 170\cdot 281^{4} + 17\cdot 281^{5} + 125\cdot 281^{6} + 58\cdot 281^{7} + 219\cdot 281^{8} +O\left(281^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 257 + 41\cdot 281 + 187\cdot 281^{2} + 209\cdot 281^{3} + 161\cdot 281^{4} + 101\cdot 281^{5} + 274\cdot 281^{6} + 40\cdot 281^{7} + 94\cdot 281^{8} +O\left(281^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 266 + 4\cdot 281 + 173\cdot 281^{2} + 168\cdot 281^{3} + 179\cdot 281^{4} + 29\cdot 281^{5} + 173\cdot 281^{6} + 228\cdot 281^{7} + 149\cdot 281^{8} +O\left(281^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 278 + 258\cdot 281 + 188\cdot 281^{2} + 127\cdot 281^{3} + 206\cdot 281^{4} + 73\cdot 281^{5} + 53\cdot 281^{6} + 209\cdot 281^{7} + 233\cdot 281^{8} +O\left(281^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(4,7)(5,6)$ |
| $(1,2)(3,8)(4,7)(5,6)$ |
| $(1,7,8,5,2,4,3,6)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,8,2,3)(4,5,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $-4$ |
| $2$ | $2$ | $(4,7)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,6)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,8,2,3)(4,5,7,6)$ | $0$ |
| $2$ | $4$ | $(1,3,2,8)(4,5,7,6)$ | $0$ |
| $4$ | $4$ | $(1,7,2,4)(3,5,8,6)$ | $0$ |
| $4$ | $8$ | $(1,6,3,4,2,5,8,7)$ | $0$ |
| $4$ | $8$ | $(1,4,3,5,2,7,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
