Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 + 103\cdot 131 + 80\cdot 131^{2} + 69\cdot 131^{3} + 64\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 94\cdot 131 + 7\cdot 131^{2} + 46\cdot 131^{3} + 44\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 99 + 106\cdot 131 + 69\cdot 131^{2} + 68\cdot 131^{3} + 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 116 + 37\cdot 131 + 33\cdot 131^{2} + 24\cdot 131^{3} + 43\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 127 + 50\cdot 131 + 70\cdot 131^{2} + 53\cdot 131^{3} + 108\cdot 131^{4} +O\left(131^{ 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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
