Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 39 + 104\cdot 109 + 99\cdot 109^{2} + 9\cdot 109^{3} + 90\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 + 32\cdot 109 + 55\cdot 109^{2} + 67\cdot 109^{3} + 53\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 52 + 93\cdot 109 + 47\cdot 109^{2} + 66\cdot 109^{3} + 43\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 + 61\cdot 109 + 25\cdot 109^{2} + 45\cdot 109^{3} + 35\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 77 + 12\cdot 109 + 20\cdot 109^{2} + 13\cdot 109^{3} + 76\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 89 + 94\cdot 109 + 100\cdot 109^{2} + 109^{3} + 11\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 92 + 74\cdot 109 + 77\cdot 109^{2} + 6\cdot 109^{3} + 14\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 95 + 70\cdot 109 + 8\cdot 109^{2} + 7\cdot 109^{3} + 3\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,8,5)(2,6,3,4)$ |
| $(1,7,8,5)(2,4,3,6)$ |
| $(1,8)(2,3)(4,6)(5,7)$ |
| $(1,6,8,4)(2,5,3,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(2,3)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,5)(4,8)$ | $0$ |
| $1$ | $4$ | $(1,7,8,5)(2,4,3,6)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,8,7)(2,6,3,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,4)(2,5,3,7)$ | $0$ |
| $2$ | $4$ | $(1,7,8,5)(2,6,3,4)$ | $0$ |
| $2$ | $4$ | $(1,2,8,3)(4,5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
