Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 25\cdot 167 + 68\cdot 167^{2} + 100\cdot 167^{3} + 32\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 20\cdot 167 + 154\cdot 167^{2} + 78\cdot 167^{3} + 97\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 + 5\cdot 167 + 26\cdot 167^{2} + 135\cdot 167^{3} + 15\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 65 + 145\cdot 167 + 157\cdot 167^{2} + 146\cdot 167^{3} + 25\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 102 + 21\cdot 167 + 9\cdot 167^{2} + 20\cdot 167^{3} + 141\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 114 + 161\cdot 167 + 140\cdot 167^{2} + 31\cdot 167^{3} + 151\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 152 + 146\cdot 167 + 12\cdot 167^{2} + 88\cdot 167^{3} + 69\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 160 + 141\cdot 167 + 98\cdot 167^{2} + 66\cdot 167^{3} + 134\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
