Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 101\cdot 109 + 38\cdot 109^{2} + 54\cdot 109^{3} + 48\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 59\cdot 109 + 23\cdot 109^{2} + 86\cdot 109^{3} + 69\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 73\cdot 109 + 31\cdot 109^{2} + 92\cdot 109^{3} + 47\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 17\cdot 109 + 37\cdot 109^{2} + 77\cdot 109^{3} + 2\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 + 93\cdot 109 + 14\cdot 109^{2} + 94\cdot 109^{3} + 51\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 86 + 98\cdot 109 + 20\cdot 109^{2} + 14\cdot 109^{3} + 44\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 94 + 90\cdot 109 + 21\cdot 109^{2} + 30\cdot 109^{3} + 18\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 105 + 10\cdot 109 + 29\cdot 109^{2} + 96\cdot 109^{3} + 43\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,6)(3,7)(4,5)$ |
| $(4,8)(6,7)$ |
| $(1,3,5,2)(4,7,8,6)$ |
| $(1,5)(2,3)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $2$ | $(4,8)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,3,5,2)(4,6,8,7)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,5,3)(4,7,8,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,5,2)(4,7,8,6)$ | $0$ |
| $2$ | $4$ | $(1,4,5,8)(2,7,3,6)$ | $0$ |
| $2$ | $4$ | $(1,7,5,6)(2,8,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
