Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 227 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 184\cdot 227 + 117\cdot 227^{2} + 136\cdot 227^{3} + 72\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 68 + 86\cdot 227 + 42\cdot 227^{2} + 224\cdot 227^{3} + 136\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 77 + 15\cdot 227 + 206\cdot 227^{2} + 25\cdot 227^{3} + 113\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 86 + 47\cdot 227 + 5\cdot 227^{2} + 196\cdot 227^{3} + 107\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 131 + 193\cdot 227 + 98\cdot 227^{2} + 173\cdot 227^{3} + 70\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 145 + 219\cdot 227 + 143\cdot 227^{2} + 101\cdot 227^{3} + 202\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 163 + 211\cdot 227 + 154\cdot 227^{2} + 109\cdot 227^{3} + 111\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 216 + 176\cdot 227 + 138\cdot 227^{2} + 167\cdot 227^{3} + 92\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,6,5,8,2,4,7)$ |
| $(1,8)(2,3)(4,6)(5,7)$ |
| $(2,5)(3,7)(4,6)$ |
| $(1,4,8,6)(2,5,3,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $-2$ |
| $4$ | $2$ | $(2,5)(3,7)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,6,8,4)(2,7,3,5)$ | $0$ |
| $2$ | $8$ | $(1,3,6,5,8,2,4,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,4,3,8,7,6,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
