Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 + 116\cdot 167 + 69\cdot 167^{2} + 139\cdot 167^{3} + 57\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 + 26\cdot 167 + 145\cdot 167^{2} + 41\cdot 167^{3} + 97\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 54 + 99\cdot 167 + 136\cdot 167^{3} + 119\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 80 + 9\cdot 167 + 76\cdot 167^{2} + 38\cdot 167^{3} + 159\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 87 + 157\cdot 167 + 90\cdot 167^{2} + 128\cdot 167^{3} + 7\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 113 + 67\cdot 167 + 166\cdot 167^{2} + 30\cdot 167^{3} + 47\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 117 + 140\cdot 167 + 21\cdot 167^{2} + 125\cdot 167^{3} + 69\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 143 + 50\cdot 167 + 97\cdot 167^{2} + 27\cdot 167^{3} + 109\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,7,6)(2,3,8,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
