Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 + 75\cdot 173 + 137\cdot 173^{2} + 169\cdot 173^{3} + 68\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 + 41\cdot 173 + 16\cdot 173^{2} + 141\cdot 173^{3} + 45\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 57 + 141\cdot 173 + 123\cdot 173^{2} + 43\cdot 173^{3} + 146\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 70 + 14\cdot 173 + 47\cdot 173^{2} + 148\cdot 173^{3} + 109\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 98 + 23\cdot 173 + 131\cdot 173^{2} + 85\cdot 173^{3} + 24\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 105 + 34\cdot 173 + 120\cdot 173^{2} + 54\cdot 173^{3} + 130\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 143 + 64\cdot 173 + 40\cdot 173^{2} + 25\cdot 173^{3} + 7\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 144 + 123\cdot 173 + 75\cdot 173^{2} + 23\cdot 173^{3} + 159\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,4,7)(2,6,5,3)$ |
| $(1,5,4,2)(3,8,6,7)$ |
| $(1,8)(2,5)(4,7)$ |
| $(1,4)(2,5)(3,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)(7,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,8)(2,5)(4,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,8,4,7)(2,6,5,3)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,5,4,2)(3,8,6,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,8,6,4,5,7,3)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,5,8,3,4,2,7,6)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
