Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 29\cdot 173 + 81\cdot 173^{2} + 33\cdot 173^{3} + 35\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 38 + 142\cdot 173 + 103\cdot 173^{2} + 122\cdot 173^{3} + 11\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 87 + 99\cdot 173 + 59\cdot 173^{2} + 167\cdot 173^{3} + 146\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 94 + 162\cdot 173 + 81\cdot 173^{2} + 68\cdot 173^{3} + 34\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 121 + 85\cdot 173 + 19\cdot 173^{2} + 127\cdot 173^{3} + 117\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
