Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 71\cdot 131 + 115\cdot 131^{2} + 127\cdot 131^{3} + 15\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 62 + 130\cdot 131 + 72\cdot 131^{2} + 112\cdot 131^{3} + 56\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 63 + 126\cdot 131 + 62\cdot 131^{2} + 45\cdot 131^{3} + 111\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 85 + 16\cdot 131 + 115\cdot 131^{2} + 129\cdot 131^{3} + 93\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 86 + 12\cdot 131 + 105\cdot 131^{2} + 62\cdot 131^{3} + 17\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 96 + 47\cdot 131 + 99\cdot 131^{2} + 89\cdot 131^{3} + 40\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 119 + 64\cdot 131 + 10\cdot 131^{2} + 107\cdot 131^{3} + 77\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 126 + 53\cdot 131 + 73\cdot 131^{2} + 110\cdot 131^{3} + 109\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,5,6)(3,7,8,4)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,2,5,6)(3,7,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
