Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 45\cdot 149 + 19\cdot 149^{2} + 12\cdot 149^{3} + 127\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 66\cdot 149 + 22\cdot 149^{2} + 90\cdot 149^{3} + 45\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 24\cdot 149 + 58\cdot 149^{2} + 63\cdot 149^{3} + 41\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 7\cdot 149 + 47\cdot 149^{2} + 43\cdot 149^{3} + 127\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 137\cdot 149 + 80\cdot 149^{2} + 15\cdot 149^{3} + 100\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 69 + 4\cdot 149 + 142\cdot 149^{2} + 43\cdot 149^{3} + 140\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 140 + 122\cdot 149 + 21\cdot 149^{2} + 134\cdot 149^{3} + 92\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 142 + 39\cdot 149 + 55\cdot 149^{2} + 44\cdot 149^{3} + 70\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,6)(3,7)(4,5)$ |
| $(2,4)(3,8)(5,7)$ |
| $(1,6)(2,8)(3,4)(5,7)$ |
| $(1,5,6,7)(2,4,8,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,8)(3,4)(5,7)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(2,4)(3,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,6,7)(2,4,8,3)$ | $0$ |
| $2$ | $8$ | $(1,3,5,2,6,4,7,8)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,2,7,3,6,8,5,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
