Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 337 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57 + 124\cdot 337 + 149\cdot 337^{2} + 116\cdot 337^{3} + 44\cdot 337^{4} + 83\cdot 337^{5} + 150\cdot 337^{6} +O\left(337^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 78 + 179\cdot 337 + 161\cdot 337^{2} + 320\cdot 337^{3} + 108\cdot 337^{4} + 326\cdot 337^{5} + 96\cdot 337^{6} +O\left(337^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 + 335\cdot 337 + 26\cdot 337^{2} + 291\cdot 337^{3} + 107\cdot 337^{4} + 22\cdot 337^{5} + 25\cdot 337^{6} +O\left(337^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 125 + 167\cdot 337 + 218\cdot 337^{2} + 271\cdot 337^{3} + 327\cdot 337^{4} + 51\cdot 337^{5} + 251\cdot 337^{6} +O\left(337^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 212 + 169\cdot 337 + 118\cdot 337^{2} + 65\cdot 337^{3} + 9\cdot 337^{4} + 285\cdot 337^{5} + 85\cdot 337^{6} +O\left(337^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 255 + 337 + 310\cdot 337^{2} + 45\cdot 337^{3} + 229\cdot 337^{4} + 314\cdot 337^{5} + 311\cdot 337^{6} +O\left(337^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 259 + 157\cdot 337 + 175\cdot 337^{2} + 16\cdot 337^{3} + 228\cdot 337^{4} + 10\cdot 337^{5} + 240\cdot 337^{6} +O\left(337^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 280 + 212\cdot 337 + 187\cdot 337^{2} + 220\cdot 337^{3} + 292\cdot 337^{4} + 253\cdot 337^{5} + 186\cdot 337^{6} +O\left(337^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,4,6)(2,5,3,8)$ |
| $(3,6)(4,5)$ |
| $(2,7)(3,6)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $4$ | $(1,7,4,6)(2,5,3,8)$ | $0$ |
| $4$ | $4$ | $(1,6,4,7)(2,8,3,5)$ | $0$ |
| $4$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(1,2,4,6)(3,8,7,5)$ | $0$ |
| $4$ | $4$ | $(1,6,4,2)(3,5,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
