Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 139\cdot 149 + 34\cdot 149^{2} + 54\cdot 149^{3} + 32\cdot 149^{4} + 111\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 124\cdot 149 + 96\cdot 149^{2} + 138\cdot 149^{3} + 37\cdot 149^{4} + 129\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 + 95\cdot 149 + 111\cdot 149^{2} + 97\cdot 149^{3} + 110\cdot 149^{4} + 75\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 + 133\cdot 149 + 21\cdot 149^{2} + 97\cdot 149^{3} + 13\cdot 149^{4} + 101\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 81 + 15\cdot 149 + 127\cdot 149^{2} + 51\cdot 149^{3} + 135\cdot 149^{4} + 47\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 108 + 53\cdot 149 + 37\cdot 149^{2} + 51\cdot 149^{3} + 38\cdot 149^{4} + 73\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 133 + 24\cdot 149 + 52\cdot 149^{2} + 10\cdot 149^{3} + 111\cdot 149^{4} + 19\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 137 + 9\cdot 149 + 114\cdot 149^{2} + 94\cdot 149^{3} + 116\cdot 149^{4} + 37\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(5,8)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $8$ | $(1,7,5,6,8,2,4,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,6,4,7,8,3,5,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
