Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 44\cdot 269 + 167\cdot 269^{2} + 68\cdot 269^{3} + 145\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 + 218\cdot 269 + 59\cdot 269^{2} + 110\cdot 269^{3} + 12\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 54 + 224\cdot 269 + 150\cdot 269^{2} + 241\cdot 269^{3} + 164\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 67 + 235\cdot 269 + 141\cdot 269^{2} + 200\cdot 269^{3} + 259\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 152 + 35\cdot 269 + 73\cdot 269^{2} + 35\cdot 269^{3} + 109\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 241 + 31\cdot 269 + 108\cdot 269^{2} + 178\cdot 269^{3} + 195\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 246 + 15\cdot 269 + 225\cdot 269^{2} + 237\cdot 269^{3} + 65\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 265 + 269 + 150\cdot 269^{2} + 3\cdot 269^{3} + 123\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,6)(4,8)(5,7)$ |
| $(1,6,3,2)(4,7,8,5)$ |
| $(2,6)(4,8)$ |
| $(1,5)(2,8)(3,7)(4,6)$ |
| $(1,3)(2,4,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,6)(4,8)(5,7)$ | $-4$ |
| $2$ | $2$ | $(1,5)(2,8)(3,7)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(2,6)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,7)(3,8)(5,6)$ | $0$ |
| $4$ | $4$ | $(1,6,3,2)(4,7,8,5)$ | $0$ |
| $4$ | $4$ | $(1,8,5,6)(2,3,4,7)$ | $0$ |
| $4$ | $4$ | $(1,6,5,8)(2,7,4,3)$ | $0$ |
| $4$ | $4$ | $(1,3)(2,4,6,8)$ | $0$ |
| $4$ | $4$ | $(1,3)(2,8,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
