Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 227 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 179\cdot 227 + 117\cdot 227^{2} + 203\cdot 227^{3} + 140\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 + 207\cdot 227 + 41\cdot 227^{2} + 82\cdot 227^{3} + 112\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 86 + 129\cdot 227 + 194\cdot 227^{2} + 146\cdot 227^{3} + 173\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 112 + 217\cdot 227 + 220\cdot 227^{2} + 166\cdot 227^{3} + 151\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 120 + 77\cdot 227 + 161\cdot 227^{2} + 128\cdot 227^{3} + 151\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 131 + 4\cdot 227 + 132\cdot 227^{2} + 183\cdot 227^{3} + 130\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 188 + 52\cdot 227 + 210\cdot 227^{2} + 126\cdot 227^{3} + 30\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 204 + 39\cdot 227 + 56\cdot 227^{2} + 96\cdot 227^{3} + 16\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,4,8)(2,6,3,7)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(1,6,4,7)(2,8,3,5)$ |
| $(1,7)(2,3)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,7)(2,3)(4,6)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,6,4,7)(2,8,3,5)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,5,4,8)(2,6,3,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,6,2,4,8,7,3)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,8,6,3,4,5,7,2)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
