Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 17\cdot 101 + 53\cdot 101^{2} + 38\cdot 101^{3} + 34\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 89\cdot 101 + 76\cdot 101^{2} + 27\cdot 101^{3} + 77\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 84\cdot 101 + 97\cdot 101^{2} + 29\cdot 101^{3} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 89\cdot 101 + 25\cdot 101^{2} + 96\cdot 101^{3} + 10\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 61 + 11\cdot 101 + 75\cdot 101^{2} + 4\cdot 101^{3} + 90\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 + 16\cdot 101 + 3\cdot 101^{2} + 71\cdot 101^{3} + 100\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 95 + 11\cdot 101 + 24\cdot 101^{2} + 73\cdot 101^{3} + 23\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 98 + 83\cdot 101 + 47\cdot 101^{2} + 62\cdot 101^{3} + 66\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,4,2,6)(3,8,5,7)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,7)(3,4)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,2,6)(3,8,5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
