Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 37\cdot 149 + 80\cdot 149^{2} + 131\cdot 149^{3} + 28\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 123\cdot 149 + 134\cdot 149^{2} + 72\cdot 149^{3} + 124\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 14\cdot 149 + 48\cdot 149^{2} + 19\cdot 149^{3} + 11\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 100\cdot 149 + 102\cdot 149^{2} + 109\cdot 149^{3} + 106\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 119 + 48\cdot 149 + 46\cdot 149^{2} + 39\cdot 149^{3} + 42\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 126 + 134\cdot 149 + 100\cdot 149^{2} + 129\cdot 149^{3} + 137\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 129 + 25\cdot 149 + 14\cdot 149^{2} + 76\cdot 149^{3} + 24\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 136 + 111\cdot 149 + 68\cdot 149^{2} + 17\cdot 149^{3} + 120\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,3)(5,6,8,7)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,4,3)(5,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
