Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 82\cdot 109 + 67\cdot 109^{2} + 26\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 48\cdot 109 + 52\cdot 109^{2} + 88\cdot 109^{3} + 101\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 97\cdot 109 + 26\cdot 109^{2} + 84\cdot 109^{3} + 23\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 108\cdot 109 + 14\cdot 109^{2} + 80\cdot 109^{3} + 55\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 + 54\cdot 109 + 82\cdot 109^{2} + 7\cdot 109^{3} + 101\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 83 + 92\cdot 109 + 99\cdot 109^{2} + 3\cdot 109^{3} + 65\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 89 + 6\cdot 109 + 6\cdot 109^{2} + 68\cdot 109^{3} + 35\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 90 + 54\cdot 109 + 85\cdot 109^{2} + 102\cdot 109^{3} + 26\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,5)(7,8)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,4,2,6)(3,7,5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(3,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,4,2,6)(3,7,5,8)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,2,4)(3,8,5,7)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,2,8)(3,4,5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,2,3)(4,8,6,7)$ | $0$ |
| $2$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
