Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 281 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 169\cdot 281 + 52\cdot 281^{2} + 40\cdot 281^{3} + 109\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 105 + 6\cdot 281 + 158\cdot 281^{2} + 152\cdot 281^{3} + 224\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 155 + 255\cdot 281 + 150\cdot 281^{2} + 247\cdot 281^{3} + 218\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 186 + 142\cdot 281 + 38\cdot 281^{2} + 237\cdot 281^{3} + 118\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 219 + 67\cdot 281 + 204\cdot 281^{2} + 235\cdot 281^{3} + 47\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 226 + 37\cdot 281 + 147\cdot 281^{2} + 133\cdot 281^{3} + 180\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 235 + 107\cdot 281 + 61\cdot 281^{2} + 133\cdot 281^{3} + 95\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 268 + 55\cdot 281 + 30\cdot 281^{2} + 225\cdot 281^{3} + 128\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(3,7,8,4)$ |
| $(1,4)(2,3)(5,7)(6,8)$ |
| $(1,5)(2,6)(3,8)(4,7)$ |
| $(1,6)(2,5)(3,7)(4,8)$ |
| $(3,8)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $-4$ |
| $2$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $0$ |
| $4$ | $4$ | $(1,3,2,7)(4,5,8,6)$ | $0$ |
| $4$ | $4$ | $(1,7,2,3)(4,6,8,5)$ | $0$ |
| $4$ | $4$ | $(1,7,5,4)(2,8,6,3)$ | $0$ |
| $4$ | $4$ | $(1,6,5,2)(3,8)$ | $0$ |
| $4$ | $4$ | $(1,2,5,6)(3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
