Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 129\cdot 173 + 45\cdot 173^{2} + 42\cdot 173^{3} + 6\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 34\cdot 173 + 73\cdot 173^{2} + 24\cdot 173^{3} + 138\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 157\cdot 173 + 26\cdot 173^{2} + 149\cdot 173^{3} + 42\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 75 + 68\cdot 173 + 129\cdot 173^{2} + 126\cdot 173^{3} + 84\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 112 + 61\cdot 173 + 20\cdot 173^{2} + 87\cdot 173^{3} + 68\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 130 + 9\cdot 173 + 21\cdot 173^{2} + 75\cdot 173^{3} + 148\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 149 + 149\cdot 173 + 165\cdot 173^{2} + 131\cdot 173^{3} + 90\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 159 + 81\cdot 173 + 36\cdot 173^{2} + 55\cdot 173^{3} + 112\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,5)(6,7)$ |
| $(1,7)(3,6)(5,8)$ |
| $(1,8)(2,4)(3,6)(5,7)$ |
| $(1,7,8,5)(2,6,4,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,7)(3,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,7,8,5)(2,6,4,3)$ | $0$ |
| $2$ | $8$ | $(1,4,7,3,8,2,5,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,5,4,8,6,7,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
