Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 + 41\cdot 131 + 34\cdot 131^{2} + 108\cdot 131^{3} + 49\cdot 131^{4} + 78\cdot 131^{5} + 97\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 66\cdot 131 + 115\cdot 131^{2} + 82\cdot 131^{3} + 59\cdot 131^{4} + 30\cdot 131^{5} + 74\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 69\cdot 131 + 78\cdot 131^{2} + 122\cdot 131^{3} + 3\cdot 131^{4} + 87\cdot 131^{5} + 88\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 116\cdot 131 + 95\cdot 131^{2} + 84\cdot 131^{3} + 112\cdot 131^{4} + 109\cdot 131^{5} + 30\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 76 + 35\cdot 131 + 78\cdot 131^{2} + 62\cdot 131^{3} + 75\cdot 131^{4} + 96\cdot 131^{5} + 111\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 83 + 7\cdot 131 + 63\cdot 131^{2} + 45\cdot 131^{3} + 69\cdot 131^{4} + 5\cdot 131^{5} + 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 102 + 40\cdot 131 + 9\cdot 131^{2} + 123\cdot 131^{3} + 69\cdot 131^{4} + 99\cdot 131^{5} + 30\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 117 + 15\cdot 131 + 49\cdot 131^{2} + 25\cdot 131^{3} + 83\cdot 131^{4} + 16\cdot 131^{5} + 89\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,2,5,6,7,8,3)$ |
| $(2,8)(3,5)$ |
| $(3,5)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,6)(2,8)(3,5)(4,7)$ |
$-4$ |
| $2$ |
$2$ |
$(3,5)(4,7)$ |
$0$ |
| $4$ |
$2$ |
$(2,8)(3,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,6,8)(3,4,5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,6,8)(3,7,5,4)$ |
$0$ |
| $4$ |
$8$ |
$(1,4,2,5,6,7,8,3)$ |
$0$ |
| $4$ |
$8$ |
$(1,5,8,4,6,3,2,7)$ |
$0$ |
| $4$ |
$8$ |
$(1,4,2,3,6,7,8,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,3,8,4,6,5,2,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
