Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 281 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 26 + 24\cdot 281 + 31\cdot 281^{2} + 101\cdot 281^{3} + 44\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 39\cdot 281 + 264\cdot 281^{2} + 179\cdot 281^{3} + 229\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 + 29\cdot 281 + 277\cdot 281^{2} + 74\cdot 281^{3} + 260\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 62 + 119\cdot 281 + 165\cdot 281^{2} + 165\cdot 281^{3} + 165\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 101 + 253\cdot 281 + 95\cdot 281^{2} + 62\cdot 281^{3} + 61\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 131 + 191\cdot 281 + 272\cdot 281^{2} + 260\cdot 281^{3} + 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 161 + 170\cdot 281 + 114\cdot 281^{2} + 31\cdot 281^{3} + 39\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 277 + 15\cdot 281 + 184\cdot 281^{2} + 247\cdot 281^{3} + 40\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,7,6)(2,4,5,8)$ |
| $(1,2)(5,7)$ |
| $(1,7)(2,5)(3,6)(4,8)$ |
| $(3,4)(6,8)$ |
| $(3,8,4,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,7)(6,8)$ |
$-4$ |
| $2$ |
$2$ |
$(1,7)(2,5)(3,6)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,2)(5,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(2,8)(3,5)(4,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,7,6)(2,4,5,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,7,3)(2,8,5,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,8,2,6)(3,5,4,7)$ |
$0$ |
| $4$ |
$4$ |
$(3,8,4,6)(5,7)$ |
$0$ |
| $4$ |
$4$ |
$(3,6,4,8)(5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
