Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 40 + 16\cdot 149 + 125\cdot 149^{2} + 121\cdot 149^{3} + 56\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 69 + 42\cdot 149 + 142\cdot 149^{2} + 31\cdot 149^{3} + 17\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 77 + 107\cdot 149 + 36\cdot 149^{2} + 45\cdot 149^{3} + 10\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 79 + 11\cdot 149 + 110\cdot 149^{2} + 112\cdot 149^{3} + 130\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 111 + 78\cdot 149 + 69\cdot 149^{2} + 31\cdot 149^{3} + 93\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 113 + 131\cdot 149 + 142\cdot 149^{2} + 98\cdot 149^{3} + 64\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 122 + 134\cdot 149 + 66\cdot 149^{2} + 45\cdot 149^{3} + 43\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 136 + 72\cdot 149 + 51\cdot 149^{2} + 108\cdot 149^{3} + 30\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,6)(4,8)(5,7)$ |
| $(1,4,2,5)(3,7,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,6)(4,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,4,2,5)(3,7,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
