Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 89\cdot 173 + 101\cdot 173^{2} + 74\cdot 173^{3} + 111\cdot 173^{4} + 115\cdot 173^{5} +O\left(173^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 96\cdot 173 + 166\cdot 173^{2} + 35\cdot 173^{3} + 47\cdot 173^{4} + 42\cdot 173^{5} +O\left(173^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 76 + 84\cdot 173^{2} + 166\cdot 173^{3} + 20\cdot 173^{4} + 25\cdot 173^{5} +O\left(173^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 78 + 7\cdot 173 + 149\cdot 173^{2} + 127\cdot 173^{3} + 129\cdot 173^{4} + 124\cdot 173^{5} +O\left(173^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 118 + 121\cdot 173 + 35\cdot 173^{2} + 61\cdot 173^{3} + 79\cdot 173^{4} + 153\cdot 173^{5} +O\left(173^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 128 + 158\cdot 173 + 67\cdot 173^{2} + 25\cdot 173^{3} + 173^{4} + 161\cdot 173^{5} +O\left(173^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 139 + 90\cdot 173 + 27\cdot 173^{2} + 118\cdot 173^{3} + 103\cdot 173^{4} + 117\cdot 173^{5} +O\left(173^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 149 + 127\cdot 173 + 59\cdot 173^{2} + 82\cdot 173^{3} + 25\cdot 173^{4} + 125\cdot 173^{5} +O\left(173^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,6)(3,7)(4,5)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(1,5,2,6,4,8,3,7)$ |
| $(1,3,4,2)(5,7,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,8)(2,6)(3,7)(4,5)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(5,6)(7,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,2)(5,7,8,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,7,3,8,4,6,2,5)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,8,2,7,4,5,3,6)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
