Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 + 86\cdot 101 + 19\cdot 101^{2} + 17\cdot 101^{3} + 55\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 86\cdot 101 + 95\cdot 101^{2} + 97\cdot 101^{3} + 57\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 + 73\cdot 101 + 12\cdot 101^{2} + 13\cdot 101^{3} + 75\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 + 9\cdot 101 + 52\cdot 101^{2} + 17\cdot 101^{3} + 35\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 74 + 32\cdot 101 + 16\cdot 101^{2} + 53\cdot 101^{3} + 36\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 81 + 19\cdot 101 + 34\cdot 101^{2} + 69\cdot 101^{3} + 53\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 94 + 92\cdot 101 + 33\cdot 101^{2} + 52\cdot 101^{3} + 42\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 99 + 2\cdot 101 + 38\cdot 101^{2} + 83\cdot 101^{3} + 47\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,6)(3,7,5,8)$ |
| $(1,3)(2,8)(4,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,5)(3,6)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,2,4,6)(3,7,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
