Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 91 + 180\cdot 269 + 41\cdot 269^{2} + 87\cdot 269^{3} + 148\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 111 + 129\cdot 269 + 187\cdot 269^{2} + 5\cdot 269^{3} + 249\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 142 + 232\cdot 269 + 95\cdot 269^{2} + 43\cdot 269^{3} + 42\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 146 + 30\cdot 269 + 183\cdot 269^{2} + 172\cdot 269^{3} + 218\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 182 + 37\cdot 269 + 134\cdot 269^{2} + 121\cdot 269^{3} + 184\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 189 + 24\cdot 269 + 33\cdot 269^{2} + 136\cdot 269^{3} + 175\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 230 + 246\cdot 269 + 226\cdot 269^{2} + 177\cdot 269^{3} + 106\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 256 + 193\cdot 269 + 173\cdot 269^{2} + 62\cdot 269^{3} + 220\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(4,5)$ |
| $(1,5,7,4)(2,8)$ |
| $(1,5)(2,6)(3,8)(4,7)$ |
| $(1,7)(2,8)(3,6)(4,5)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,8)(4,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,4)(5,8)(6,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,5,2)(4,8,7,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,5,3)(4,6,7,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,7,4)(2,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,7,5)(2,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,7,8)(3,5,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
