Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 86\cdot 269 + 126\cdot 269^{2} + 86\cdot 269^{3} + 113\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 45\cdot 269 + 147\cdot 269^{2} + 169\cdot 269^{3} + 210\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 88 + 143\cdot 269 + 113\cdot 269^{2} + 253\cdot 269^{3} + 189\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 159 + 153\cdot 269 + 192\cdot 269^{2} + 200\cdot 269^{3} + 155\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 176 + 7\cdot 269 + 85\cdot 269^{2} + 110\cdot 269^{3} + 125\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 188 + 144\cdot 269 + 110\cdot 269^{2} + 177\cdot 269^{3} + 174\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 210 + 264\cdot 269 + 30\cdot 269^{2} + 135\cdot 269^{3} + 151\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 216 + 230\cdot 269 + 212\cdot 269^{3} + 223\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,5,6)(2,8,7,4)$ |
| $(1,4,3,2,5,8,6,7)$ |
| $(2,7)(4,8)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(2,7)(4,8)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,3,5,6)(2,8,7,4)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,6,5,3)(2,4,7,8)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,3,5,6)(2,4,7,8)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,3,2,5,8,6,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,6,4,5,7,3,8)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,6,7,5,8,3,2)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,7,3,4,5,2,6,8)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
