Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 281 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 59 + 20\cdot 281 + 4\cdot 281^{2} + 109\cdot 281^{3} + 26\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 94 + 154\cdot 281 + 133\cdot 281^{2} + 251\cdot 281^{3} + 160\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 125 + 163\cdot 281 + 152\cdot 281^{2} + 22\cdot 281^{3} + 108\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 142 + 113\cdot 281 + 139\cdot 281^{2} + 240\cdot 281^{3} + 199\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 151 + 253\cdot 281 + 31\cdot 281^{2} + 135\cdot 281^{3} + 244\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 152 + 105\cdot 281 + 86\cdot 281^{2} + 4\cdot 281^{3} + 177\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 172 + 245\cdot 281 + 234\cdot 281^{2} + 212\cdot 281^{3} + 125\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 230 + 67\cdot 281 + 60\cdot 281^{2} + 148\cdot 281^{3} + 81\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,2,3,8,5,4,7)$ |
| $(1,2,8,4)(3,5,7,6)$ |
| $(1,8)(2,4)(3,7)(5,6)$ |
| $(3,7)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,4)(3,7)(5,6)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(3,7)(5,6)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,2,8,4)(3,5,7,6)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,4,8,2)(3,6,7,5)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,2,8,4)(3,6,7,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,2,3,8,5,4,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,4,6,8,7,2,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,4,7,8,5,2,3)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,7,2,6,8,3,4,5)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
