Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 + 106\cdot 131 + 70\cdot 131^{2} + 55\cdot 131^{3} + 102\cdot 131^{4} + 6\cdot 131^{5} + 96\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 + 14\cdot 131 + 32\cdot 131^{2} + 83\cdot 131^{3} + 97\cdot 131^{4} + 87\cdot 131^{5} + 51\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 + 3\cdot 131 + 108\cdot 131^{2} + 3\cdot 131^{3} + 38\cdot 131^{4} + 22\cdot 131^{5} + 4\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 63 + 45\cdot 131 + 57\cdot 131^{2} + 44\cdot 131^{3} + 4\cdot 131^{4} + 53\cdot 131^{5} + 71\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 + 85\cdot 131 + 73\cdot 131^{2} + 86\cdot 131^{3} + 126\cdot 131^{4} + 77\cdot 131^{5} + 59\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 70 + 127\cdot 131 + 22\cdot 131^{2} + 127\cdot 131^{3} + 92\cdot 131^{4} + 108\cdot 131^{5} + 126\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 98 + 116\cdot 131 + 98\cdot 131^{2} + 47\cdot 131^{3} + 33\cdot 131^{4} + 43\cdot 131^{5} + 79\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 103 + 24\cdot 131 + 60\cdot 131^{2} + 75\cdot 131^{3} + 28\cdot 131^{4} + 124\cdot 131^{5} + 34\cdot 131^{6} +O\left(131^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,4,2,3,8,5,7,6)$ |
| $(1,8)(2,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,8)(2,7)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,2,3,8,5,7,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,7,4,8,6,2,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,2,4,8,3,7,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,7,6,8,5,2,3)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
