Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 28\cdot 101 + 74\cdot 101^{2} + 33\cdot 101^{3} + 90\cdot 101^{4} + 25\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 85\cdot 101 + 76\cdot 101^{2} + 57\cdot 101^{3} + 72\cdot 101^{4} + 7\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 + 39\cdot 101 + 40\cdot 101^{2} + 45\cdot 101^{3} + 100\cdot 101^{4} + 28\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 96\cdot 101 + 42\cdot 101^{2} + 69\cdot 101^{3} + 82\cdot 101^{4} + 10\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 4\cdot 101 + 58\cdot 101^{2} + 31\cdot 101^{3} + 18\cdot 101^{4} + 90\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 61\cdot 101 + 60\cdot 101^{2} + 55\cdot 101^{3} + 72\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 96 + 15\cdot 101 + 24\cdot 101^{2} + 43\cdot 101^{3} + 28\cdot 101^{4} + 93\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 100 + 72\cdot 101 + 26\cdot 101^{2} + 67\cdot 101^{3} + 10\cdot 101^{4} + 75\cdot 101^{5} +O\left(101^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,6,4)(2,3,5,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
