Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 89\cdot 101 + 48\cdot 101^{2} + 46\cdot 101^{3} + 67\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 60\cdot 101 + 82\cdot 101^{2} + 12\cdot 101^{3} + 57\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 + 29\cdot 101 + 52\cdot 101^{2} + 76\cdot 101^{3} + 33\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 86\cdot 101^{2} + 42\cdot 101^{3} + 23\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 100\cdot 101 + 14\cdot 101^{2} + 58\cdot 101^{3} + 77\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 71\cdot 101 + 48\cdot 101^{2} + 24\cdot 101^{3} + 67\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 85 + 40\cdot 101 + 18\cdot 101^{2} + 88\cdot 101^{3} + 43\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 91 + 11\cdot 101 + 52\cdot 101^{2} + 54\cdot 101^{3} + 33\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,7,4,6)(2,5,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
