Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 229 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 211\cdot 229 + 126\cdot 229^{2} + 52\cdot 229^{3} + 123\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 85 + 35\cdot 229 + 90\cdot 229^{2} + 205\cdot 229^{3} + 6\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 150 + 57\cdot 229 + 188\cdot 229^{2} + 58\cdot 229^{3} + 201\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 153 + 153\cdot 229 + 8\cdot 229^{2} + 180\cdot 229^{3} + 124\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 171 + 119\cdot 229 + 103\cdot 229^{2} + 173\cdot 229^{3} + 64\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 178 + 120\cdot 229 + 200\cdot 229^{2} + 27\cdot 229^{3} + 111\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 194 + 89\cdot 229 + 32\cdot 229^{2} + 159\cdot 229^{3} + 216\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 211 + 127\cdot 229 + 165\cdot 229^{2} + 58\cdot 229^{3} + 67\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,5)(2,8,7,6)$ |
| $(1,7,4,2)(3,8,5,6)$ |
| $(1,4)(3,8)(5,6)$ |
| $(1,4)(2,7)(3,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,4)(3,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,7,4,2)(3,8,5,6)$ | $0$ |
| $4$ | $4$ | $(1,3,4,5)(2,8,7,6)$ | $0$ |
| $2$ | $8$ | $(1,8,7,5,4,6,2,3)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,6,7,3,4,8,2,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
