Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 52 + 6\cdot 173 + 98\cdot 173^{2} + 51\cdot 173^{3} + 4\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 89 + 73\cdot 173 + 154\cdot 173^{2} + 95\cdot 173^{3} + 70\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 92 + 74\cdot 173 + 146\cdot 173^{2} + 127\cdot 173^{3} + 90\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 113 + 18\cdot 173 + 120\cdot 173^{2} + 70\cdot 173^{3} + 7\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
