Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 271 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 90 + 160\cdot 271 + 188\cdot 271^{2} + 65\cdot 271^{3} + 264\cdot 271^{4} + 138\cdot 271^{5} +O\left(271^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 108 + 196\cdot 271 + 211\cdot 271^{2} + 101\cdot 271^{3} + 91\cdot 271^{4} + 248\cdot 271^{5} +O\left(271^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 114 + 171\cdot 271 + 50\cdot 271^{2} + 228\cdot 271^{3} + 48\cdot 271^{4} + 107\cdot 271^{5} +O\left(271^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 138 + 7\cdot 271 + 53\cdot 271^{2} + 180\cdot 271^{3} + 202\cdot 271^{4} + 87\cdot 271^{5} +O\left(271^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 190 + 9\cdot 271 + 77\cdot 271^{2} + 141\cdot 271^{3} + 117\cdot 271^{4} + 247\cdot 271^{5} +O\left(271^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 231 + 13\cdot 271 + 91\cdot 271^{2} + 146\cdot 271^{3} + 137\cdot 271^{4} + 47\cdot 271^{5} +O\left(271^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 242 + 40\cdot 271 + 245\cdot 271^{2} + 52\cdot 271^{3} + 122\cdot 271^{4} + 49\cdot 271^{5} +O\left(271^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 244 + 212\cdot 271 + 166\cdot 271^{2} + 167\cdot 271^{3} + 99\cdot 271^{4} + 157\cdot 271^{5} +O\left(271^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(5,7)$ |
| $(4,8)(5,7)$ |
| $(3,6)(5,7)$ |
| $(1,8,3,5,2,4,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $-4$ |
| $2$ | $2$ | $(4,8)(5,7)$ | $0$ |
| $4$ | $2$ | $(3,6)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,3,2,6)(4,7,8,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,6)(4,5,8,7)$ | $0$ |
| $4$ | $8$ | $(1,8,3,5,2,4,6,7)$ | $0$ |
| $4$ | $8$ | $(1,5,6,8,2,7,3,4)$ | $0$ |
| $4$ | $8$ | $(1,8,3,7,2,4,6,5)$ | $0$ |
| $4$ | $8$ | $(1,7,6,8,2,5,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
