Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 62\cdot 101 + 25\cdot 101^{2} + 83\cdot 101^{3} + 10\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 96\cdot 101 + 21\cdot 101^{2} + 88\cdot 101^{3} + 3\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 31\cdot 101 + 11\cdot 101^{2} + 15\cdot 101^{3} + 95\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 + 16\cdot 101 + 28\cdot 101^{2} + 7\cdot 101^{3} + 64\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 + 64\cdot 101 + 60\cdot 101^{2} + 19\cdot 101^{3} + 96\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 89\cdot 101 + 59\cdot 101^{3} + 47\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 73 + 33\cdot 101 + 46\cdot 101^{2} + 52\cdot 101^{3} + 79\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 89 + 9\cdot 101 + 7\cdot 101^{2} + 79\cdot 101^{3} + 6\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,7,6)(2,3,8,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
