Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 + 252\cdot 337 + 134\cdot 337^{2} + 322\cdot 337^{3} + 278\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 + 237\cdot 337 + 117\cdot 337^{2} + 327\cdot 337^{3} + 273\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 57 + 328\cdot 337 + 318\cdot 337^{2} + 244\cdot 337^{3} + 112\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 62 + 16\cdot 337 + 259\cdot 337^{2} + 284\cdot 337^{3} + 185\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 169 + 234\cdot 337 + 122\cdot 337^{2} + 107\cdot 337^{3} + 119\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 179 + 24\cdot 337 + 326\cdot 337^{2} + 29\cdot 337^{3} + 44\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 195 + 181\cdot 337 + 198\cdot 337^{2} + 209\cdot 337^{3} + 32\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 276 + 73\cdot 337 + 207\cdot 337^{2} + 158\cdot 337^{3} + 300\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,8,3)(2,6,7,4)$ |
| $(1,2,8,7)(3,6,5,4)$ |
| $(1,8)(2,7)(3,5)(4,6)$ |
| $(1,7)(2,8)(4,6)$ |
| $(1,8)(2,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $-4$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,6,5,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,6,5,4)$ | $0$ |
| $4$ | $4$ | $(1,5,8,3)(2,6,7,4)$ | $0$ |
| $4$ | $8$ | $(1,5,2,4,8,3,7,6)$ | $0$ |
| $4$ | $8$ | $(1,3,7,4,8,5,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
