Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 84\cdot 109 + 49\cdot 109^{2} + 42\cdot 109^{3} + 38\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 63\cdot 109 + 17\cdot 109^{2} + 87\cdot 109^{3} + 75\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 81\cdot 109 + 92\cdot 109^{2} + 9\cdot 109^{3} + 93\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 67 + 2\cdot 109 + 59\cdot 109^{2} + 90\cdot 109^{3} + 10\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 73 + 70\cdot 109 + 42\cdot 109^{2} + 104\cdot 109^{3} + 2\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 75 + 102\cdot 109 + 35\cdot 109^{2} + 67\cdot 109^{3} + 85\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 90 + 37\cdot 109 + 107\cdot 109^{2} + 36\cdot 109^{3} + 35\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 99 + 102\cdot 109 + 30\cdot 109^{2} + 106\cdot 109^{3} + 93\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,5)(3,8)(6,7)$ |
| $(2,5)(6,7)$ |
| $(1,6,4,7)(2,3,5,8)$ |
| $(1,3,4,8)(2,6,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(2,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $0$ |
| $1$ | $4$ | $(1,3,4,8)(2,6,5,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,4,3)(2,7,5,6)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,4,7)(2,3,5,8)$ | $0$ |
| $2$ | $4$ | $(1,3,4,8)(2,7,5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,4,2)(3,7,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
