Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 87\cdot 131 + 44\cdot 131^{2} + 26\cdot 131^{3} + 31\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 95\cdot 131 + 73\cdot 131^{2} + 102\cdot 131^{3} + 92\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 74\cdot 131 + 38\cdot 131^{2} + 59\cdot 131^{3} + 16\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 6\cdot 131 + 8\cdot 131^{2} + 110\cdot 131^{3} + 108\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 + 84\cdot 131 + 31\cdot 131^{2} + 10\cdot 131^{3} + 83\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 62 + 117\cdot 131 + 125\cdot 131^{2} + 116\cdot 131^{3} + 13\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 87 + 9\cdot 131 + 103\cdot 131^{2} + 88\cdot 131^{3} + 39\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 125 + 49\cdot 131 + 98\cdot 131^{2} + 9\cdot 131^{3} + 7\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,6)(3,8)(4,5)$ |
| $(1,5)(2,8)(3,6)(4,7)$ |
| $(1,3)(4,5)(7,8)$ |
| $(1,3,7,8)(2,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,8)(3,6)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,3,7,8)(2,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,4,8,2,7,5,3,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,2,3,4,7,6,8,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
