Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 43 + 20\cdot 337 + 242\cdot 337^{2} + 138\cdot 337^{3} + 166\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 133 + 11\cdot 337 + 213\cdot 337^{2} + 166\cdot 337^{3} + 105\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 172 + 49\cdot 337 + 56\cdot 337^{2} + 201\cdot 337^{3} + 193\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 214 + 267\cdot 337 + 270\cdot 337^{2} + 87\cdot 337^{3} + 229\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 258 + 291\cdot 337 + 143\cdot 337^{2} + 243\cdot 337^{3} + 224\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 279 + 173\cdot 337 + 30\cdot 337^{2} + 183\cdot 337^{3} + 326\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 289 + 287\cdot 337 + 305\cdot 337^{2} + 285\cdot 337^{3} + 275\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 300 + 245\cdot 337 + 85\cdot 337^{2} + 41\cdot 337^{3} + 163\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,6,7,5,3,4)$ |
| $(2,5)(4,6)$ |
| $(1,3,7,8)(2,4,5,6)$ |
| $(1,7)(2,6)(4,5)$ |
| $(1,7)(2,5)(3,8)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $-4$ |
| $2$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,7)(3,4)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,8,7,3)(2,6,5,4)$ | $0$ |
| $2$ | $4$ | $(1,3,7,8)(2,6,5,4)$ | $0$ |
| $4$ | $4$ | $(1,2,7,5)(3,6,8,4)$ | $0$ |
| $4$ | $8$ | $(1,2,8,6,7,5,3,4)$ | $0$ |
| $4$ | $8$ | $(1,5,3,6,7,2,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
