Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 110\cdot 173 + 82\cdot 173^{2} + 73\cdot 173^{3} + 62\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 114\cdot 173 + 100\cdot 173^{2} + 123\cdot 173^{3} + 153\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 171\cdot 173 + 73\cdot 173^{2} + 145\cdot 173^{3} + 93\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 61\cdot 173 + 17\cdot 173^{2} + 63\cdot 173^{3} + 3\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 16\cdot 173 + 28\cdot 173^{2} + 122\cdot 173^{3} + 51\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 77 + 43\cdot 173 + 74\cdot 173^{2} + 119\cdot 173^{3} + 103\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 90 + 29\cdot 173 + 5\cdot 173^{2} + 128\cdot 173^{3} + 131\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 101 + 146\cdot 173 + 136\cdot 173^{2} + 89\cdot 173^{3} + 91\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,7)(3,8)(4,6)$ |
| $(1,5)(2,6)(3,8)(4,7)$ |
| $(1,7)(2,3)(4,5)(6,8)$ |
| $(1,8,5,3)(2,4,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,8)(4,7)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,7)(2,3)(4,5)(6,8)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(2,7)(3,8)(4,6)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,8,5,3)(2,4,6,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,8,4,5,6,3,7)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,4,3,2,5,7,8,6)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
