Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 146\cdot 167 + 105\cdot 167^{2} + 32\cdot 167^{3} + 59\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 17\cdot 167 + 131\cdot 167^{2} + 25\cdot 167^{3} + 79\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 57\cdot 167 + 26\cdot 167^{2} + 120\cdot 167^{3} + 3\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 49\cdot 167 + 147\cdot 167^{2} + 156\cdot 167^{3} + 161\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 146\cdot 167 + 78\cdot 167^{2} + 26\cdot 167^{3} + 94\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 68 + 97\cdot 167 + 146\cdot 167^{2} + 148\cdot 167^{3} + 87\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 76 + 33\cdot 167 + 55\cdot 167^{2} + 32\cdot 167^{3} + 16\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 94 + 121\cdot 167 + 143\cdot 167^{2} + 124\cdot 167^{3} + 165\cdot 167^{4} +O\left(167^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(3,8)(5,6)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,7,3,6)(2,4,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
