Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 129\cdot 131 + 3\cdot 131^{2} + 118\cdot 131^{3} + 70\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 112\cdot 131 + 72\cdot 131^{2} + 124\cdot 131^{3} + 36\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 14\cdot 131 + 37\cdot 131^{2} + 104\cdot 131^{3} + 5\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 + 75\cdot 131 + 38\cdot 131^{2} + 30\cdot 131^{3} + 25\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 72 + 104\cdot 131 + 34\cdot 131^{2} + 104\cdot 131^{3} + 68\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 86 + 16\cdot 131 + 87\cdot 131^{2} + 94\cdot 131^{3} + 60\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 114 + 85\cdot 131 + 91\cdot 131^{2} + 84\cdot 131^{3} + 14\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 124 + 116\cdot 131 + 26\cdot 131^{2} + 125\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 |
| $(1,5,7,6)(2,8,4,3)$ |
| $(2,8)(3,4)(5,6)$ |
| $(1,2,6,3,7,4,5,8)$ |
| $(1,7)(2,4)(3,8)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $-2$ |
| $4$ | $2$ | $(2,8)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,6,7,5)(2,3,4,8)$ | $0$ |
| $2$ | $8$ | $(1,2,6,3,7,4,5,8)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,5,2,7,8,6,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
