Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53\cdot 101 + 43\cdot 101^{2} + 82\cdot 101^{3} + 77\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 57\cdot 101 + 61\cdot 101^{2} + 65\cdot 101^{3} + 91\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 19\cdot 101 + 56\cdot 101^{2} + 96\cdot 101^{3} + 71\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 44\cdot 101 + 39\cdot 101^{2} + 52\cdot 101^{3} + 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 + 75\cdot 101 + 99\cdot 101^{2} + 60\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 16\cdot 101 + 98\cdot 101^{2} + 52\cdot 101^{3} + 73\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 66 + 42\cdot 101 + 71\cdot 101^{2} + 69\cdot 101^{3} + 94\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 86 + 95\cdot 101 + 34\cdot 101^{2} + 84\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 |
| $(1,2,6,5)(3,4,8,7)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,6,5)(3,4,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
