Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 229 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 33\cdot 229 + 177\cdot 229^{2} + 46\cdot 229^{3} + 101\cdot 229^{4} + 19\cdot 229^{5} + 201\cdot 229^{6} + 174\cdot 229^{7} +O\left(229^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 68 + 54\cdot 229 + 64\cdot 229^{2} + 182\cdot 229^{3} + 23\cdot 229^{4} + 204\cdot 229^{5} + 190\cdot 229^{6} + 210\cdot 229^{7} +O\left(229^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 95 + 196\cdot 229 + 130\cdot 229^{2} + 10\cdot 229^{3} + 208\cdot 229^{4} + 96\cdot 229^{5} + 181\cdot 229^{6} + 165\cdot 229^{7} +O\left(229^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 112 + 213\cdot 229 + 163\cdot 229^{2} + 149\cdot 229^{3} + 117\cdot 229^{4} + 108\cdot 229^{5} + 172\cdot 229^{6} + 161\cdot 229^{7} +O\left(229^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 117 + 15\cdot 229 + 65\cdot 229^{2} + 79\cdot 229^{3} + 111\cdot 229^{4} + 120\cdot 229^{5} + 56\cdot 229^{6} + 67\cdot 229^{7} +O\left(229^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 134 + 32\cdot 229 + 98\cdot 229^{2} + 218\cdot 229^{3} + 20\cdot 229^{4} + 132\cdot 229^{5} + 47\cdot 229^{6} + 63\cdot 229^{7} +O\left(229^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 161 + 174\cdot 229 + 164\cdot 229^{2} + 46\cdot 229^{3} + 205\cdot 229^{4} + 24\cdot 229^{5} + 38\cdot 229^{6} + 18\cdot 229^{7} +O\left(229^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 217 + 195\cdot 229 + 51\cdot 229^{2} + 182\cdot 229^{3} + 127\cdot 229^{4} + 209\cdot 229^{5} + 27\cdot 229^{6} + 54\cdot 229^{7} +O\left(229^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(2,6,7,3)$ |
| $(1,6,5,7,8,3,4,2)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(2,7)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(2,6,7,3)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,3,7,6)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,4,8,5)(2,7)(3,6)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,5,8,4)(2,7)(3,6)$ | $-\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $4$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $0$ |
| $4$ | $8$ | $(1,7,4,6,8,2,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
