Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 44\cdot 131 + 39\cdot 131^{2} + 51\cdot 131^{3} + 81\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 4\cdot 131 + 86\cdot 131^{2} + 114\cdot 131^{3} + 107\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 115\cdot 131 + 69\cdot 131^{2} + 75\cdot 131^{3} + 74\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 45\cdot 131 + 43\cdot 131^{2} + 2\cdot 131^{3} + 129\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 + 116\cdot 131 + 73\cdot 131^{2} + 26\cdot 131^{3} + 122\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 + 97\cdot 131 + 62\cdot 131^{2} + 69\cdot 131^{3} + 81\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 79 + 99\cdot 131 + 70\cdot 131^{2} + 102\cdot 131^{3} + 45\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 123 + 131 + 78\cdot 131^{2} + 81\cdot 131^{3} + 12\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,7)(5,6)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,8)(2,3,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
