Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 337 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 39 + 167\cdot 337 + 300\cdot 337^{2} + 155\cdot 337^{3} + 102\cdot 337^{4} + 263\cdot 337^{5} + 321\cdot 337^{6} + 330\cdot 337^{7} +O\left(337^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 93 + 125\cdot 337 + 309\cdot 337^{2} + 203\cdot 337^{3} + 206\cdot 337^{4} + 112\cdot 337^{5} + 143\cdot 337^{6} + 166\cdot 337^{7} +O\left(337^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 96 + 256\cdot 337 + 179\cdot 337^{2} + 104\cdot 337^{3} + 76\cdot 337^{4} + 296\cdot 337^{5} + 308\cdot 337^{6} + 176\cdot 337^{7} +O\left(337^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 98 + 193\cdot 337 + 292\cdot 337^{2} + 258\cdot 337^{3} + 285\cdot 337^{4} + 93\cdot 337^{5} + 73\cdot 337^{6} + 196\cdot 337^{7} +O\left(337^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 239 + 143\cdot 337 + 44\cdot 337^{2} + 78\cdot 337^{3} + 51\cdot 337^{4} + 243\cdot 337^{5} + 263\cdot 337^{6} + 140\cdot 337^{7} +O\left(337^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 241 + 80\cdot 337 + 157\cdot 337^{2} + 232\cdot 337^{3} + 260\cdot 337^{4} + 40\cdot 337^{5} + 28\cdot 337^{6} + 160\cdot 337^{7} +O\left(337^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 244 + 211\cdot 337 + 27\cdot 337^{2} + 133\cdot 337^{3} + 130\cdot 337^{4} + 224\cdot 337^{5} + 193\cdot 337^{6} + 170\cdot 337^{7} +O\left(337^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 298 + 169\cdot 337 + 36\cdot 337^{2} + 181\cdot 337^{3} + 234\cdot 337^{4} + 73\cdot 337^{5} + 15\cdot 337^{6} + 6\cdot 337^{7} +O\left(337^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(5,8)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,8)(4,5)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,5,8,4)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,8)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,7)(5,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,7)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,8,4)(2,3,7,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,8,6)(2,4,7,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,3,4,2,8,6,5,7)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,4,3,8,2,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
